text:
section\<open>Arities of internalized formulas\<close>
theory Arities
  imports FrecR
begin

lemma arity_upair_fm : "\<lbrakk>  t1\<in>nat ; t2\<in>nat ; up\<in>nat  \<rbrakk> \<Longrightarrow>
  arity(upair_fm(t1,t2,up)) = \<Union> {succ(t1),succ(t2),succ(up)}"
  unfolding  upair_fm_def
  using nat_union_abs1 nat_union_abs2 pred_Un
  by auto

lemma arity_pair_fm : "\<lbrakk>  t1\<in>nat ; t2\<in>nat ; p\<in>nat  \<rbrakk> \<Longrightarrow>
  arity(pair_fm(t1,t2,p)) = \<Union> {succ(t1),succ(t2),succ(p)}"
  unfolding pair_fm_def
  using arity_upair_fm nat_union_abs1 nat_union_abs2 pred_Un
  by auto

lemma arity_composition_fm :
  "\<lbrakk> r\<in>nat ; s\<in>nat ; t\<in>nat \<rbrakk> \<Longrightarrow> arity(composition_fm(r,s,t)) = \<Union> {succ(r), succ(s), succ(t)}"
  unfolding composition_fm_def
  using arity_pair_fm nat_union_abs1 nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_domain_fm :
    "\<lbrakk> r\<in>nat ; z\<in>nat \<rbrakk> \<Longrightarrow> arity(domain_fm(r,z)) = succ(r) \<union> succ(z)"
  unfolding domain_fm_def
  using arity_pair_fm nat_union_abs1 nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_range_fm :
    "\<lbrakk> r\<in>nat ; z\<in>nat \<rbrakk> \<Longrightarrow> arity(range_fm(r,z)) = succ(r) \<union> succ(z)"
  unfolding range_fm_def
  using arity_pair_fm nat_union_abs1 nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_union_fm :
  "\<lbrakk> x\<in>nat ; y\<in>nat ; z\<in>nat \<rbrakk> \<Longrightarrow> arity(union_fm(x,y,z)) = \<Union> {succ(x), succ(y), succ(z)}"
  unfolding union_fm_def
  using  nat_union_abs1 nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_image_fm :
  "\<lbrakk> x\<in>nat ; y\<in>nat ; z\<in>nat \<rbrakk> \<Longrightarrow> arity(image_fm(x,y,z)) = \<Union> {succ(x), succ(y), succ(z)}"
  unfolding image_fm_def
  using arity_pair_fm  nat_union_abs1 nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_pre_image_fm :
  "\<lbrakk> x\<in>nat ; y\<in>nat ; z\<in>nat \<rbrakk> \<Longrightarrow> arity(pre_image_fm(x,y,z)) = \<Union> {succ(x), succ(y), succ(z)}"
  unfolding pre_image_fm_def
  using arity_pair_fm  nat_union_abs1 nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_big_union_fm :
  "\<lbrakk> x\<in>nat ; y\<in>nat \<rbrakk> \<Longrightarrow> arity(big_union_fm(x,y)) = succ(x) \<union> succ(y)"
  unfolding big_union_fm_def
  using nat_union_abs1 nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_fun_apply_fm :
  "\<lbrakk> x\<in>nat ; y\<in>nat ; f\<in>nat \<rbrakk> \<Longrightarrow>
    arity(fun_apply_fm(f,x,y)) =  succ(f) \<union> succ(x) \<union> succ(y)"
  unfolding fun_apply_fm_def
  using arity_upair_fm arity_image_fm arity_big_union_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_field_fm :
    "\<lbrakk> r\<in>nat ; z\<in>nat \<rbrakk> \<Longrightarrow> arity(field_fm(r,z)) = succ(r) \<union> succ(z)"
  unfolding field_fm_def
  using arity_pair_fm arity_domain_fm arity_range_fm arity_union_fm
    nat_union_abs1 nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_empty_fm :
    "\<lbrakk> r\<in>nat \<rbrakk> \<Longrightarrow> arity(empty_fm(r)) = succ(r)"
  unfolding empty_fm_def
  using nat_union_abs1 nat_union_abs2 pred_Un_distrib
  by simp

lemma arity_succ_fm :
  "\<lbrakk>x\<in>nat;y\<in>nat\<rbrakk> \<Longrightarrow> arity(succ_fm(x,y)) = succ(x) \<union> succ(y)"
  unfolding succ_fm_def cons_fm_def
  using arity_upair_fm arity_union_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma number1arity__fm :
    "\<lbrakk> r\<in>nat \<rbrakk> \<Longrightarrow> arity(number1_fm(r)) = succ(r)"
  unfolding number1_fm_def
  using arity_empty_fm arity_succ_fm nat_union_abs1 nat_union_abs2 pred_Un_distrib
  by simp

lemma arity_function_fm :
    "\<lbrakk> r\<in>nat \<rbrakk> \<Longrightarrow> arity(function_fm(r)) = succ(r)"
  unfolding function_fm_def
  using arity_pair_fm nat_union_abs1 nat_union_abs2 pred_Un_distrib
  by simp

lemma arity_relation_fm :
    "\<lbrakk> r\<in>nat \<rbrakk> \<Longrightarrow> arity(relation_fm(r)) = succ(r)"
  unfolding relation_fm_def
  using arity_pair_fm nat_union_abs1 nat_union_abs2 pred_Un_distrib
  by simp

lemma arity_restriction_fm :
    "\<lbrakk> r\<in>nat ; z\<in>nat ; A\<in>nat \<rbrakk> \<Longrightarrow> arity(restriction_fm(A,z,r)) = succ(A) \<union> succ(r) \<union> succ(z)"
  unfolding restriction_fm_def
  using arity_pair_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_typed_function_fm :
  "\<lbrakk> x\<in>nat ; y\<in>nat ; f\<in>nat \<rbrakk> \<Longrightarrow>
    arity(typed_function_fm(f,x,y)) = \<Union> {succ(f), succ(x), succ(y)}"
  unfolding typed_function_fm_def
  using arity_pair_fm arity_relation_fm arity_function_fm arity_domain_fm
    nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_subset_fm :
  "\<lbrakk>x\<in>nat ; y\<in>nat\<rbrakk> \<Longrightarrow> arity(subset_fm(x,y)) = succ(x) \<union> succ(y)"
  unfolding subset_fm_def
  using nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_transset_fm :
  "\<lbrakk>x\<in>nat\<rbrakk> \<Longrightarrow> arity(transset_fm(x)) = succ(x)"
  unfolding transset_fm_def
  using arity_subset_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_ordinal_fm :
  "\<lbrakk>x\<in>nat\<rbrakk> \<Longrightarrow> arity(ordinal_fm(x)) = succ(x)"
  unfolding ordinal_fm_def
  using arity_transset_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_limit_ordinal_fm :
  "\<lbrakk>x\<in>nat\<rbrakk> \<Longrightarrow> arity(limit_ordinal_fm(x)) = succ(x)"
  unfolding limit_ordinal_fm_def
  using arity_ordinal_fm arity_succ_fm arity_empty_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_finite_ordinal_fm :
  "\<lbrakk>x\<in>nat\<rbrakk> \<Longrightarrow> arity(finite_ordinal_fm(x)) = succ(x)"
  unfolding finite_ordinal_fm_def
  using arity_ordinal_fm arity_limit_ordinal_fm arity_succ_fm arity_empty_fm
    nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_omega_fm :
  "\<lbrakk>x\<in>nat\<rbrakk> \<Longrightarrow> arity(omega_fm(x)) = succ(x)"
  unfolding omega_fm_def
  using arity_limit_ordinal_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_cartprod_fm :
  "\<lbrakk> A\<in>nat ; B\<in>nat ; z\<in>nat \<rbrakk> \<Longrightarrow> arity(cartprod_fm(A,B,z)) = succ(A) \<union> succ(B) \<union> succ(z)"
  unfolding cartprod_fm_def
  using arity_pair_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_fst_fm :
  "\<lbrakk>x\<in>nat ; t\<in>nat\<rbrakk> \<Longrightarrow> arity(fst_fm(x,t)) = succ(x) \<union> succ(t)"
  unfolding fst_fm_def
  using arity_pair_fm arity_empty_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_snd_fm :
  "\<lbrakk>x\<in>nat ; t\<in>nat\<rbrakk> \<Longrightarrow> arity(snd_fm(x,t)) = succ(x) \<union> succ(t)"
  unfolding snd_fm_def
  using arity_pair_fm arity_empty_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_snd_snd_fm :
  "\<lbrakk>x\<in>nat ; t\<in>nat\<rbrakk> \<Longrightarrow> arity(snd_snd_fm(x,t)) = succ(x) \<union> succ(t)"
  unfolding snd_snd_fm_def hcomp_fm_def
  using arity_snd_fm arity_empty_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_ftype_fm :
  "\<lbrakk>x\<in>nat ; t\<in>nat\<rbrakk> \<Longrightarrow> arity(ftype_fm(x,t)) = succ(x) \<union> succ(t)"
  unfolding ftype_fm_def
  using arity_fst_fm
  by auto

lemma name1arity__fm :
  "\<lbrakk>x\<in>nat ; t\<in>nat\<rbrakk> \<Longrightarrow> arity(name1_fm(x,t)) = succ(x) \<union> succ(t)"
  unfolding name1_fm_def hcomp_fm_def
  using arity_fst_fm arity_snd_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma name2arity__fm :
  "\<lbrakk>x\<in>nat ; t\<in>nat\<rbrakk> \<Longrightarrow> arity(name2_fm(x,t)) = succ(x) \<union> succ(t)"
  unfolding name2_fm_def hcomp_fm_def
  using arity_fst_fm arity_snd_snd_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_cond_of_fm :
  "\<lbrakk>x\<in>nat ; t\<in>nat\<rbrakk> \<Longrightarrow> arity(cond_of_fm(x,t)) = succ(x) \<union> succ(t)"
  unfolding cond_of_fm_def hcomp_fm_def
  using arity_snd_fm arity_snd_snd_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_singleton_fm :
  "\<lbrakk>x\<in>nat ; t\<in>nat\<rbrakk> \<Longrightarrow> arity(singleton_fm(x,t)) = succ(x) \<union> succ(t)"
  unfolding singleton_fm_def cons_fm_def
  using arity_union_fm arity_upair_fm arity_empty_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_Memrel_fm :
  "\<lbrakk>x\<in>nat ; t\<in>nat\<rbrakk> \<Longrightarrow> arity(Memrel_fm(x,t)) = succ(x) \<union> succ(t)"
  unfolding Memrel_fm_def
  using  arity_pair_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_quasinat_fm :
  "\<lbrakk>x\<in>nat\<rbrakk> \<Longrightarrow> arity(quasinat_fm(x)) = succ(x)"
  unfolding quasinat_fm_def cons_fm_def
  using arity_succ_fm arity_empty_fm
    nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_is_recfun_fm :
  "\<lbrakk>p\<in>formula ; v\<in>nat ; n\<in>nat; Z\<in>nat;i\<in>nat\<rbrakk> \<Longrightarrow>  arity(p) = i \<Longrightarrow>
  arity(is_recfun_fm(p,v,n,Z)) = succ(v) \<union> succ(n) \<union> succ(Z) \<union> pred(pred(pred(pred(i))))"
  unfolding is_recfun_fm_def
  using arity_upair_fm arity_pair_fm arity_pre_image_fm arity_restriction_fm
    nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_is_wfrec_fm :
  "\<lbrakk>p\<in>formula ; v\<in>nat ; n\<in>nat; Z\<in>nat ; i\<in>nat\<rbrakk> \<Longrightarrow> arity(p) = i \<Longrightarrow>
    arity(is_wfrec_fm(p,v,n,Z)) = succ(v) \<union> succ(n) \<union> succ(Z) \<union> pred(pred(pred(pred(pred(i)))))"
  unfolding is_wfrec_fm_def
  using arity_succ_fm  arity_is_recfun_fm
     nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_is_nat_case_fm :
  "\<lbrakk>p\<in>formula ; v\<in>nat ; n\<in>nat; Z\<in>nat; i\<in>nat\<rbrakk> \<Longrightarrow> arity(p) = i \<Longrightarrow>
    arity(is_nat_case_fm(v,p,n,Z)) = succ(v) \<union> succ(n) \<union> succ(Z) \<union> pred(pred(i))"
  unfolding is_nat_case_fm_def
  using arity_succ_fm arity_empty_fm arity_quasinat_fm
    nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_iterates_MH_fm :
  assumes "isF\<in>formula" "v\<in>nat" "n\<in>nat" "g\<in>nat" "z\<in>nat" "i\<in>nat"
      "arity(isF) = i"
    shows "arity(iterates_MH_fm(isF,v,n,g,z)) =
           succ(v) \<union> succ(n) \<union> succ(g) \<union> succ(z) \<union> pred(pred(pred(pred(i))))"
proof -
  let ?\<phi> = "Exists(And(fun_apply_fm(succ(succ(succ(g))), 2, 0), Forall(Implies(Equal(0, 2), isF))))"
  let ?ar = "succ(succ(succ(g))) \<union> pred(pred(i))"
  from assms
  have "arity(?\<phi>) =?ar" "?\<phi>\<in>formula"
    using arity_fun_apply_fm
    nat_union_abs1 nat_union_abs2 pred_Un_distrib succ_Un_distrib Un_assoc[symmetric]
    by simp_all
  then
  show ?thesis
    unfolding iterates_MH_fm_def
    using arity_is_nat_case_fm[OF \<open>?\<phi>\<in>_\<close> _ _ _ _ \<open>arity(?\<phi>) = _\<close>] assms pred_succ_eq pred_Un_distrib
    by auto
qed

lemma arity_is_iterates_fm :
  assumes "p\<in>formula" "v\<in>nat" "n\<in>nat" "Z\<in>nat" "i\<in>nat"
    "arity(p) = i"
  shows "arity(is_iterates_fm(p,v,n,Z)) = succ(v) \<union> succ(n) \<union> succ(Z) \<union>
          pred(pred(pred(pred(pred(pred(pred(pred(pred(pred(pred(i)))))))))))"
proof -
  let ?\<phi> = "iterates_MH_fm(p, 7#+v, 2, 1, 0)"
  let ?\<psi> = "is_wfrec_fm(?\<phi>, 0, succ(succ(n)),succ(succ(Z)))"
  from \<open>v\<in>_\<close>
  have "arity(?\<phi>) = (8#+v) \<union> pred(pred(pred(pred(i))))" "?\<phi>\<in>formula"
    using assms arity_iterates_MH_fm nat_union_abs2
    by simp_all
  then
  have "arity(?\<psi>) = succ(succ(succ(n))) \<union> succ(succ(succ(Z))) \<union> (3#+v) \<union>
      pred(pred(pred(pred(pred(pred(pred(pred(pred(i)))))))))"
    using assms arity_is_wfrec_fm[OF \<open>?\<phi>\<in>_\<close> _ _ _ _ \<open>arity(?\<phi>) = _\<close>] nat_union_abs1 pred_Un_distrib
    by auto
  then
  show ?thesis
    unfolding is_iterates_fm_def
    using arity_Memrel_fm arity_succ_fm assms nat_union_abs1 pred_Un_distrib
    by auto
qed

lemma arity_eclose_n_fm :
  assumes "A\<in>nat" "x\<in>nat" "t\<in>nat"
  shows "arity(eclose_n_fm(A,x,t)) = succ(A) \<union> succ(x) \<union> succ(t)"
proof -
  let ?\<phi> = "big_union_fm(1,0)"
  have "arity(?\<phi>) = 2" "?\<phi>\<in>formula"
    using arity_big_union_fm nat_union_abs2
    by simp_all
  with assms
  show ?thesis
    unfolding eclose_n_fm_def
    using arity_is_iterates_fm[OF \<open>?\<phi>\<in>_\<close> _ _ _,of _ _ _ 2]
    by auto
qed

lemma arity_mem_eclose_fm :
  assumes "x\<in>nat" "t\<in>nat"
  shows "arity(mem_eclose_fm(x,t)) = succ(x) \<union> succ(t)"
proof -
  let ?\<phi>="eclose_n_fm(x #+ 2, 1, 0)"
  from \<open>x\<in>nat\<close>
  have "arity(?\<phi>) = x#+3"
    using arity_eclose_n_fm nat_union_abs2
    by simp
  with assms
  show ?thesis
    unfolding mem_eclose_fm_def
    using arity_finite_ordinal_fm nat_union_abs2 pred_Un_distrib
    by simp
qed

lemma arity_is_eclose_fm :
  "\<lbrakk>x\<in>nat ; t\<in>nat\<rbrakk> \<Longrightarrow> arity(is_eclose_fm(x,t)) = succ(x) \<union> succ(t)"
  unfolding is_eclose_fm_def
  using arity_mem_eclose_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma eclose_n1arity__fm :
  "\<lbrakk>x\<in>nat ; t\<in>nat\<rbrakk> \<Longrightarrow> arity(eclose_n1_fm(x,t)) = succ(x) \<union> succ(t)"
  unfolding eclose_n1_fm_def
  using arity_is_eclose_fm arity_singleton_fm name1arity__fm nat_union_abs2 pred_Un_distrib
  by auto

lemma eclose_n2arity__fm :
  "\<lbrakk>x\<in>nat ; t\<in>nat\<rbrakk> \<Longrightarrow> arity(eclose_n2_fm(x,t)) = succ(x) \<union> succ(t)"
  unfolding eclose_n2_fm_def
  using arity_is_eclose_fm arity_singleton_fm name2arity__fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_ecloseN_fm :
  "\<lbrakk>x\<in>nat ; t\<in>nat\<rbrakk> \<Longrightarrow> arity(ecloseN_fm(x,t)) = succ(x) \<union> succ(t)"
  unfolding ecloseN_fm_def
  using eclose_n1arity__fm eclose_n2arity__fm arity_union_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_frecR_fm :
  "\<lbrakk>a\<in>nat;b\<in>nat\<rbrakk> \<Longrightarrow> arity(frecR_fm(a,b)) = succ(a) \<union> succ(b)"
  unfolding frecR_fm_def
  using arity_ftype_fm name1arity__fm name2arity__fm arity_domain_fm
      number1arity__fm arity_empty_fm nat_union_abs2 pred_Un_distrib
  by auto

lemma arity_Collect_fm :
  assumes "x \<in> nat" "y \<in> nat" "p\<in>formula"
  shows "arity(Collect_fm(x,p,y)) = succ(x) \<union> succ(y) \<union> pred(arity(p))"
  unfolding Collect_fm_def
  using assms pred_Un_distrib
  by auto

end

subset_name: curated


TITLE: Improper Integration of A Non-even Non-odd Function From $0$ to $\infty$
QUESTION [2 upvotes]: I am trying to calculate the integral:
$$\int_{0}^\infty  \frac{x^2dx}{1+x^7}$$
I used to face this type of integration with even integrand, but the function here is not even nor odd!
Is there a trick to do it?

REPLY [1 votes]: Another way to do it is to integrate
$$
f(z) = \frac{z^2\operatorname{Log}_n z}{1+z^7}
$$
around a keyhole contour. (Here $\operatorname{Log}_n z$ is the natural branch of the complex logarithm.)
See for example this for details. (There will be a lot of work doing this for this particular integrals though, with seven poles to take care of.)


TITLE: $\psi (m)\leq \phi (m)$ or $\psi (m) \geq \phi (m)$ when $\psi (m)\neq 0$?
QUESTION [0 upvotes]: (This is different than If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic)
I was trying to solve this:

Let $G$ be a finite abelian group of order $n$ for which the number of solutions of $x^m=e$ is at most $m$ for any $m$ dividing $n$. Prove that $G$ must be cyclic. [Hint:Let $\psi (m)$ be the number of elements in $G$ of order $m$. Show that $\psi (m)\leq \phi (m)$, and use $n=\Sigma_{m\mid n}\phi (m) $]

What I have proven is that $\Sigma_{d|n}(ψ(d)−ϕ(d))=0$, then if the statement in the hint holds, ψ(d)=ϕ(d) for every d|n. n divides itself, therefore, $\psi (n)=ϕ(n)≠0$. Therefore G admits a generator, and we'll be done.The problem is to prove the statement in the hint. Yet I have conluded a different seemingly contradictory result, which doesn't help solve the problem at all, since it could be that $ψ(n)=0,ψ(n)−ψ(d)<0$ and $ψ(d)−ϕ(d)>0$ for some $d≠n$. Of course I don't know how to show that $\psi (m)\leq \phi (m)$, and here is how I got this contradiction:
If $\psi (m)\neq 0$, then let $m\neq 1,m\mid n$. Consider an element $a\in G,|a|=m$. Within its cyclic group, there will be $\phi (m)$ elements with order $m$ since any element $a^i$ in it is a generator if and only if $\gcd(m,i)=1$. Therefore $\psi(m) \geq \phi (m)$ since some element out side the cyclic group may have order $m$ too.
If this reasoning is correct, I think it will contradict with the fact in the hint which is $\phi(m) \geq \phi(m)$. The number of solutions is $\Sigma_{d|m}\psi (d)$, which is always greater than $\psi (m)$. So no matter how great $\psi (m)$ is, or how $\psi (m) >\phi (m)$, the number of solution may still be less than $m$. What's going wrong? How to prove the statment in the hint? Help me please. Also I have read through If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic, but for the accepted answer, it doesn't show how to prove the fact in the hint, and other answers do not use the fact in the hint.

REPLY [2 votes]: Let $d\mid n$ and suppose that $g\in G$ has order $d$.  Then
$$1,g,g^2,\ldots,g^{d-1}$$
are all different, and are solutions of the equation $x^d=e$ in $G$.  We have exhibited $d$ solutions of the equation, and so by assumption there are no further solutions.  Moreover, not all the powers $g^k$ have order $d$, as some will have smaller order.  In fact, $g^k$ has order $d$ if and only if $k$ is relatively prime to $d$, and so the number of elements of order $d$ is $\phi(d)$, and we have $\psi(d)=\phi(d)$.  This argument rests on the assumption that there is some element of order $d$; but if there is not then $\psi(d)=0$, and certainly in either case we have $\psi(d)\le\phi(d)$.


TITLE: What happens as the stable orbital velocity approaches the speed of light?
QUESTION [2 upvotes]: Based on my understanding of the relationship between planetary mass, orbital radius and the velocity for stable orbit, a satellite orbiting a mass equivalent to Earth with an altitude of ~5mm would have an orbital velocity of $c$.
Clearly this is a classical equation so what would actually happen if you tried to do this? Or does physics conveniently make this a non-question - whereby 5mm is inside the event horizon of an Earth mass, for instance?

REPLY [3 votes]: If you want a complete and consistent picture, I would recommend studying geodesics (orbits) in some black-hole solution of equations of GR (for reasons of simplicity, take the Schwarzschild metric). Almost all work is already done for you in this wikipedia article.
Short answer: no, the speed would never reach or exceed the speed of light. The aspects of particle dynamics and corrections to the gravitational potential from General Relativity wouldn't allow that to happen.


TITLE: Multivariable Application of inverse function theorem
QUESTION [1 upvotes]: Consider the transformation $x=G(u)$  where $x=(u_{1},u_{2},u_{3})$ given by
$x_{1} = u_{1} +(u_{3})^{2}$
$x_{2} = u_{3} -(u_{1})^{2}$
$x_{3} = u_{1} +u_{2} + u_{3}$
1) compute the derivative of this transformation.
i am having a bit of trouble understanding what this all means
i know that i am supposed to be Computing Df from the other example in my textbook but the way $u_{1}$ and $x_{1}$ are defined has me completely baffled.
2) Use the inverse function theorem to show that the transformation is locally invertable if $u_{1}u_{2} > 0$
Could someone perhaps layout the inverse function theorem in a more palatable way to this question? my textbook gives only a proof of the theorem and im not sure how to apply it.

REPLY [1 votes]: I suspect that you have an error in the definition of $G$ or the invertibility condition.
Let $G_1(u) = u_{1} +(u_{3})^{2}$,
 $G_{2}(u) = u_{3} -(u_{1})^{2}$,
$G_{3}(u) = u_{1} +u_{2} + u_{3}$.
Compute $DG(u) = \begin{bmatrix}
\frac{\partial G_1(u)}{\partial u_1} & \frac{\partial G_1(u)}{\partial u_2} & \frac{\partial G_1(u)}{\partial u_3} \\
\frac{\partial G_2(u)}{\partial u_1} & \frac{\partial G_2(u)}{\partial u_2} & \frac{\partial G_2(u)}{\partial u_3} \\
\frac{\partial G_3(u)}{\partial u_1} & \frac{\partial G_3(u)}{\partial u_2} & \frac{\partial G_3(u)}{\partial u_3}
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 & 2 u_3 \\
-2 u_1 & 0 & 1 \\
1 & 1 & 1
\end{bmatrix}
$.
Computing the determinant gives $\det( DG(u)) = -1 - 4 u_1 u_3$, hence $DG(u)$ is invertible iff $u_1 u_3 \neq - \frac{1}{4}$.


TITLE: odds/probability of single prize raffle
QUESTION [0 upvotes]: this might seem basic but for all my searching i haven’t seen it said for single prize.
if the scenario is:
1 prize
100 tickets
player 1: 1 ticket
player 2: 1 ticket
player 3: 5 tickets
player 4: 20 tickets
player 5: 73 tickets
player 1 has 1/100 odds of winning and that’s unaffected by player 5 buying 73 tickets.
but wouldn’t players buying substantially more tickets change the possible outcome for player 1? regardless of the odds remaining the same?
appreciate any helps thanks!

REPLY [0 votes]: If all tickets are sold, then Player 1 will have a $1\%%$ chance to win the prize. The distribution of the other tickets doesn't matter.
You may have seen a phrase in some sweepstakes stating: "The odds of winning depends on the number of entries received." Therefore, if only $20$ tickets are sold, then Player 1's odds of winning are $1/20$. Or if a thousand tickets are sold, Player 1's odds are $1/1000$, and so forth.


text:
\begin{document}

\date{\today}

\title[Pointwise convergence along a curve]{Pointwise convergence along a tangential curve for the fractional Schr\"odinger equation}

\keywords{Fractional Schr\"odinger equation, pointwise convergence, fractional dimension}

\subjclass[2010]{35Q41}

\begin{abstract}
In this paper we study the pointwise convergence problem along a tangential curve for the fractional Schr\"odinger equations in one spatial dimension and estimate the capacitary dimension of the divergence set. We extend a prior paper by Lee and the first author for the classical Schr\"odinger equation, which in itself contains a result due to Lee, Vargas and the first author, to the fractional Schr\"odinger equation. The proof is based on a decomposition argument without time localization, which has recently been introduced by the second author.
\end{abstract}

\maketitle

\section{Introduction}
Let $d\ge1$. We consider the fractional Schr\"odinger equation on $\mathbb R^d\times \mathbb R$
\[
\begin{cases}
	i\partial_t u + (-\Delta)^{\frac{m}2} u = 0, &\\
	u(\cdot,0) = f \in H^{s}(\mathbb R^d),  &
\end{cases}
\]
with the initial data $f\in H^s(\mathbb R^d)$ and $m>0$. Here, $H^s(\mathbb R^d)$ denotes the Sobolev space of order $s$ whose norm is given by
\[
\|f\|_{H^s(\mathbb{R}^d)}=\|(1-\Delta)^\frac s2f\|_{L^2(\mathbb{R}^d)}.
\]
Then, the solution can be formally written as
\[
u(x,t) = e^{it(-\Delta)^{\frac m2}}f(x) =\left( \frac1{2\pi}\right)^d \int_{\mathbb R^d} e^{i(x\cdot \xi + t|\xi|^m)} \widehat{f}(\xi)\,\d\xi,
\]
where $\widehat \cdot$ is the Fourier transform defined by $\widehat{f}(\xi)=\int_{\mathbb R^d} e^{-ix\cdot\xi} f(x)\,\d x$. While the classical and standard Schr\"odinger equation when $m=2$ has drawn attention in numerous fields, the fractional Sch\"odinger equation with general $m$ has also been found to be influential in recent years. In fact, it is not only a model case of a general dispersive equation \cite{CS88, KPV91} but also one of the fundamental equations in quantum mechanics \cite{Ls00, Ls02}. Since then, it has become an important subject studied by a number of authors from a variety of perspectives, see for example \cite{CKS16, CL14, GH11, GLNY18, GY14, HS15, IP14, Ke12, Ps09}.

A fundamental question is the following pointwise convergence problem of determining the minimal exponent $s$ for which
\begin{equation}\label{conv}
\lim_{t\to0} e^{it(-\Delta)^{\frac m2}}f(\gamma(x,t))=f(x)\quad\ae
\end{equation}
for $f\in H^s(\mathbb R^d)$.
Here, $\gamma$ is a continuous function such that
\[
\gamma: \mathbb R^d \times[-1,1]\to\mathbb R^d, \quad\gamma(x,0)=x.
\]
The key quality of $\gamma$ for \eqref{conv} is the way of convergence whether \textit{tangential} or \textit{non-tangential} to the hyperplane $\mathbb R^d\times\{0\}$.

The simplest example of $\gamma$ is $\gamma(x,t)=x$ and may be considered as the prototypical non-tangential case. For this $\gamma$, \eqref{conv} reduces to the seminal pointwise convergence problem, so-called Carleson's problem, originating in \cite{Cr80}. It turns out for $d=1$ and $m=2$ that \eqref{conv} holds for all $f\in H^s(\mathbb R)$ if and only if $s\ge\frac14$, due to \cite{Cr80} and Dahlberg and Kenig \cite{DK82}. Later, Sj\"olin \cite{Sj87} generalized their results and proved $s\ge\frac14$ is also necessary and sufficient even for $m>1$. In higher dimensions, $d\ge2$, the problem has attracted significant attention and been studied by many authors. When $m=2$, it has recently been understood that $s\ge\frac{d}{2(d+1)}$ is necessary for \eqref{conv} by Bourgain in \cite{Br16} and $s>\frac{d}{2(d+1)}$ is sufficient for \eqref{conv} by Du, Guth and Li \cite{DGL17} and Du and Zhang \cite{DZ18}. Ko and the first author \cite{CK18} also proved $s>\frac{d}{2(d+1)}$ is sufficient for \eqref{conv} when $m>1$ as well. The reader may also refer to in particular the work of Vega \cite{Vg88}, Lee \cite{Lee06}, Bourgain \cite{Br13}, and Du, Guth, Li and Zhang \cite{DGLZ18} as papers which have played an important role in earlier developments.

In the study of pointwise convergence problem for the Schr\"odinger equation with harmonic oscillator potential, Lee and Rogers \cite{LeeR12}
showed that any $\gamma\in C^1(\mathbb R^d\times[-1,1]\to\mathbb R^d)$, such as $\gamma(x,t)=x-(t^\kappa,0,\cdots,0)$ with $\kappa\ge1$, is essentially equivalent to the vertical line in the context of pointwise convergence problem of $\eqref{conv}$.\\

\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[scale=0.8]

\tikzset{->-/.style={decoration={
  markings,
  mark=at position .5 with {\arrow{stealth}}},postaction={decorate}}}

  \fill [fill=lightgray!30!] (2.5,0)--(5,3.5)--(0,3.5)--(2.5,0);

 \draw [->](-1,0)--(5,0);
 \draw [->](0,-0.5)--(0,4);

 \draw [color=red,thick,->-] (-0.5,3) to [out =-80, in=170] (2.5,0) ;
 \draw [color=green!80!black!, thick,->-](2.5,3)--(2.5,0);
  \draw [color=blue,thick, ->-] (0.5,3) to [out =-20, in=100] (2.5,0);

 \node [below]at (2.5,0){$x$};
 \node [below]at (5,0){$y$};
 \node [left]at (0,4){$t$};

\end{tikzpicture}
\caption{Three typical paths of convergence} \label{fig:curves}
\end{center}
\end{figure}

In contrast to the non-tangential case above, here we consider the tangential case.
The curve $\gamma$ is said to satisfy H\"older condition of order $\kappa\in(0,1]$ in $t$ if
\begin{equation}\label{cond:Holder}
|\gamma(x,t)-\gamma(x,t')|\le C_1|t-t'|^{\kappa},\quad x\in\mathbb R^d, \quad t, t'\in[-1,1]
\end{equation}
and be bilipschitz in $x$ if
\begin{equation}\label{cond:bilipschitz}
\frac{1}{C_2}|x-x'|\le|\gamma(x,t)-\gamma(x',t)|\le C_2|x-x'|,\quad t\in[-1,1],\quad x, x'\in\mathbb R^d
\end{equation}
for some $C_1, C_2 > 0$.
Then, we denote by $\Gamma(d,\kappa)$ the collection of such curves, namely,
\[
\Gamma(d,\kappa)=\{\gamma: \mathbb R^d\times[-1,1]\to\mathbb R^d:\text{$\gamma$ satisfies $\gamma(x,0)=x$, \eqref{cond:Holder} and \eqref{cond:bilipschitz}}\}.
\]
Note that $\Gamma(d,\kappa)$ contains $\gamma(x,t)=x-(t^\kappa,0,\cdots,0)$ with $0\le\kappa\le 1$. (See also Figure \ref{fig:curves}.) As a tangential case for $\gamma\in\Gamma(1,\kappa)$, Lee, Vargas and the first author \cite{CLV12} observed the crucial difference in nature from the non-tangential case; $s>\max\{\frac14,\frac{1-2\kappa}{2}\}$ is the sharp sufficient condition for \eqref{conv}.

We further consider a refinement of this question: quantifying  the sets on which the convergence \eqref{conv} fails in more precise way than Lebesgue measure. Let $0<\alpha\le d$. A positive Borel measure $\mu$ is said to be $\alpha$-dimensional if there exists a  constant $c$ such that
\begin{equation}\label{alpha-dimensional}
\mu(\mathbb B(x,r))\le c r^\alpha,
\end{equation}
where $\mathbb B(x,r)$ is the ball centered at $x\in\mathbb R^d$ with radius $r>0$.
For $f \in H^{s}(\mathbb R^d)$ and $\gamma \in \Gamma(d,\kappa)$, let us define the \textit{divergence set} by
\[
\mathfrak{D}(\gamma,f)=\{x\in \mathbb R^d:e^{it(-\Delta)^{m/2}}f(\gamma(x,t))\not\to f(x)\ {\rm as}\ t\to0\},
\]
and for a subset $X$ in $\mathbb R^d$, we also define the \textit{capacitary dimension} by
\[
\dim_c(X)=\sup\{\alpha: \mathcal M^\alpha(X)\not=\emptyset\},
\]
where $\mathcal M^\alpha(X)=\{\mu:\text{$\mu$ is $\alpha$-dimensional and $0<\mu(X)<\infty$}\}$. By the forthcoming Frostman's lemma, note that for a Borel set $X$ there exists $\mu\in\mathcal M^\alpha(X)$ if and only if the Hausdorff dimension of $X$ is greater than or equal to $\alpha$. In this case, the capacitary dimension of $X$ coincides with the Hausdorff dimension (see also \cite{Mt95}). Sj\"ogren and Sj\"olin \cite{SS89} considered this refined version of Carleson's problem for $\gamma(x,t)=x$ and $m\ge2$. Later, Barcel\'o, Bennett, Carbery and Rogers \cite{BBCR11} extended their result to $m>1$ and also obtained the sharp bound of the Hausdorff dimension of the divergence set by combining with work of \v Zubrini\'c \cite{Zb02}. In higher dimension, there are still some gaps remaining for which we refer the readers to contributions by Bennett and Rogers \cite{BR12}, Luc\`{a} and Rogers \cite{LR17, LR19b, LR19}, and aforementioned papers \cite{DGL17, DZ18, DGLZ18}. For $\gamma\in \Gamma(d,\kappa)$ and $m=2$, Lee and the first author obtained the capacitary dimension of the divergence set in \cite{CL14} as a refinement of aforementioned result by them with Vargas\footnote{
The result due to \cite{CLV12} coincides with the case $\alpha=1$ in \cite{CL14}. Consequently,
\[
|\mathfrak{D}(\gamma,f)| = 0
\] for all $f \in H^{s}(\mathbb R)$ with $s>\max\left\{\frac14,\frac12-\kappa\right\}$, where $|\cdot|$ is the Lebesgue measure defined in $\mathbb R$.
}.\\

Let $d=1$ and write $\Gamma(\kappa)=\Gamma(1,\kappa)$ in the rest of the paper. Our goal of the present note is to estimate the capacitary dimension of divergence sets for the pointwise convergence to the fractional Schr\"odinger equation along the curve $\gamma\in\Gamma(\kappa)$. Let us define evolution operator $S_t$ on appropriate input functions by
\[
S_tf(x)=\frac{1}{2\pi}\int_{\mathbb{R}}e^{i(x\xi+t|\xi|^m)}\widehat{f}(\xi)\,\d\xi.
\]
Our main results are the following.

\begin{theorem}\label{pointwise convergence with mu}
Let $m>1$, $0<\kappa\le1$, $\mu$ be an $\alpha$-dimensional measure and $\gamma\in\Gamma(\kappa)$. If $s>\max\left\{\frac14, \frac{1-\alpha}{2}, \frac{1-m\alpha\kappa}{2} \right\}$, then
\[
\lim_{t\to0}S_t(f(\gamma(x,t)))=f(x),\quad\text{$\mu$-$\ae\ x$}
\]
for all $f\in H^s(\mathbb{R})$.
\end{theorem}

By a standard argument, this is reduced to the following local maximal estimate.

\begin{theorem}\label{thm:maximal estimate with mu}
Let $m>1$, $0<\kappa\le1$, $\mu$ be an $\alpha$-dimensional measure and $\gamma\in\Gamma(\kappa)$. If $s>\max\left\{\frac14, \frac{1-\alpha}{2}, \frac{1-m\alpha\kappa}{2} \right\}$, then there exists a constant $C$ such that
\begin{equation}\label{ineq:maximal estimate with mu}
\left(\int_{-1}^1\sup_{t\in [-1,1]}|S_tf(\gamma(\cdot,t))|^2\,\d\mu(x)\right)^\frac12\le C \|f\|_{H^s}
\end{equation}
for all $f\in H^s(\mathbb{R})$.
\end{theorem}

It is straightforward to obtain some results of the above type for $m\not=2$ (more specifically $m\ge2$) by appropriately modifying the argument in \cite{CLV12} or \cite{DN19}, however, as the second author observed in his study of a related problem in a different setting \cite{Sh19}, there are some barriers with such an approach to treat $m$ near $1$. In particular, building on the ideas in \cite{Sh19} in which we completely avoid time localization techniques, we are able to handle the full range of $m>1$ and give us the sharp sufficient conditions. Here, saying sharp is meant by that: Suppose $s<\max\left\{\frac14, \frac{1-\alpha}{2}, \frac{1-m\alpha\kappa}{2} \right\}$, then there exists $\gamma\in\Gamma(\kappa)$, $\alpha$-dimensional $\mu$ and $f\in H^s(\mathbb R)$ such that \eqref{pointwise convergence with mu} fails.

Since the counterexamples can be provided by adjusting the corresponding well-known constructions (for instance, \cite{CLV12} and \cite{Sj87}) without any major difficulty, we rather focus on the sufficient conditions.
As corollaries of Theorem \ref{pointwise convergence with mu}, we have the following.

\begin{corollary}\label{divergence set estimate}
Let $m>1$, $0<\kappa\le1$, $\gamma\in\Gamma(\kappa)$. If $s>\frac14$, then
\[
\dim_c(\mathfrak{D}(\gamma,f))\le\max\left\{1-2s,\frac{1-2s}{m\kappa}\right\}.
\]
\end{corollary}

The special case when $\mu$ is the ($1$-dimensional) Lebesgue measure extends the result in \cite{CLV12} from $m=2$ to $m>1$ as follows. Here, note that the required regularity on $s$ for \eqref{conv} depends not only on $\kappa$ but $m$ as well.

\begin{corollary}\label{pointwise convergence without mu}
Let $m>1$, $0<\kappa\le1$ and $\gamma\in\Gamma(\kappa)$.  If $s>\max\{\frac14,\frac{1-m\kappa}{2}\}$, then
\begin{equation}\label{ineq:maximal estimate without mu}
\left(\int_{-1}^1\sup_{t\in [-1,1]}|S_tf(\gamma(\cdot,t))|^2\,\d x\right)^\frac12\le C \|f\|_{H^s}
\end{equation}

holds for all $f\in H^s(\mathbb{R})$.
\end{corollary}

Ding and Niu have also considered \eqref{ineq:maximal estimate without mu} for $m\ge2$ in \cite{DN19} and claim that the sharp sufficient condition is $s>\max\{\frac14,\frac{1-2\kappa}{2}\}$ (in particular, their condition is independent of $m$). Unfortunately, it appears that the arguments in \cite{DN19} are not complete and the sharp regularity threshold is  $\max\{\frac14,\frac{1-m\kappa}{2}\}$; we note that the necessity of $s\ge\max\{\frac14,\frac{1-m\kappa}{2}\}$ for \eqref{ineq:maximal estimate without mu} follows from Section \ref{sec:counterexamples} in the present paper by taking $\alpha=1$.

Combining Corollary \ref{pointwise convergence without mu} with the result from \cite{Sh19}, the results in \cite{CLV12} have been completely extended from the standard Schr\"odinger equation to the fractional Schr\"odinger equation with\footnote{
One can also consider the fractional Schr\"odinger equation with $0<m\le 1$ but the nature appears to be certainly different. For instance, see \cite{Cw82, RV08, Wl95} for $\gamma(x,t)=x$.
}
$m>1$.

\begin{remark}
Although Theorem \ref{thm:maximal estimate with mu} is stated for the fractional Schr\"odinger evolution operator, by simply following the provided proof in Section \ref{sec:proof of maximal estimate} the same conclusion is valid for a wider class of evolution operators such as
\[
S_t^{\Phi}f(x)=\frac{1}{2\pi}\int_{\mathbb{R}}e^{i(x\xi+t\Phi(\xi))}\widehat{f}(\xi)\,\d\xi,
\]
where $\Phi:\mathbb{R}\to\mathbb{R}$ is a $C^2$-function for which there exist constants $C_3,C_4>0$ such that
\begin{equation}\label{Phi'' is bounded below}
|\xi|^{2-m}\left|\frac{\d^2}{\d\xi^2}\Phi(\xi)\right|\ge C_3
\quad\text{and}\quad
|\xi|\left|\frac{\d^2}{\d\xi^2}\Phi(\xi)\right|\ge C_4\left|\frac{\d}{\d\xi}\Phi(\xi)\right|
\end{equation}
for all $|\xi|\ge1$. This class trivially contains $|\xi|^m$ whenever $m>1$.
\end{remark}

Throughout the paper, we denote $I=[-1,1]$, $A\gtrsim B$ if $A\ge CB$, $A\lesssim B$ if $A\le CB$ and $A\sim B$ if $C^{-1}B\le A \le CB$ for some constant $C>0$. The domain of certain norms is sometimes abbreviated, for its meaning is clear from the context. In the following Section \ref{sec:preliminaries}, we present useful lemmas, then first prove Theorem \ref{pointwise convergence with mu} and Corollary \ref{divergence set estimate} in Section \ref{sec:reduction arguments}. In Section \ref{sec:proof of maximal estimate}, we prove Theorem \ref{thm:maximal estimate with mu} by employing the philosophy in \cite{Sh19} of a decomposition argument without time localization. Finally, in Section \ref{sec:counterexamples} we see the shaprness of Theorem \ref{thm:maximal estimate with mu}.

\section{Preliminaries}\label{sec:preliminaries}

In this section, as we have informed, we introduce useful lemmas which we use multiple times in the rest of the paper.
\begin{lemma}[Frostman's lemma]\label{lem:Frostman}
Let $d\ge1$ and $X$ be a Borel set in $\mathbb{R}^d$. Then, $\dim_c(X)\ge\alpha$ if and only if there exists $\alpha$-dimensional measure such that $\supp \mu\subset X$ and $0<\mu(\mathbb{R}^d)<\infty$. Further, $\mu(X)>0$.
\end{lemma}
For a proof of Lemma \ref{lem:Frostman}, we refer the reader to \cite{Mt95}.
We need the following lemmas in order to prove Theorem \ref{thm:maximal estimate with mu} in Section \ref{sec:proof of maximal estimate}.

\begin{lemma}[van der Corput's lemma]\label{lem:van der Corput}
Let $-\infty<a<b<\infty$, $\phi$ be a sufficiently smooth real-valued function and $\psi$ be a bounded smooth complex-valued function. Suppose we have $|\frac{\d^k}{\d \xi^k}\phi(\xi)|\geq1$ for all $\xi \in [a,b]$. If $k=1$ and $\frac{\d}{\d \xi}\phi(\xi)$ is monotonic on $(a,b)$, or simply $k\geq2$, then there exists a constant $C_k$ such that
\[
\left|\int_a^be^{i\lambda\phi(\xi)}\psi(\xi)\,\d\xi\right|\leq C_k\lambda^{-\frac1k}\left(\int_a^b\left|\frac{\d}{\d\xi}\psi(\xi)\right|\,\d\xi+\|\psi\|_{L^{\infty}}\right)
\]
for all $\lambda > 0$.
\end{lemma}
For a proof of Lemma \ref{lem:van der Corput}, we refer the reader to \cite{St94}.

\begin{lemma}\label{lem:Young and HLS-type}
Let $0<\alpha\le1$ and $\mu$ be an $\alpha$-dimensional measure. There exists a constant $C$ such that for any interval $[a,b]$ ($-\infty<a,b<\infty$)
\begin{align}\label{ineq:Young special case}
&\left|\iint\iint g(x,t)h(x',t')\chi_{[a,b]}(x-x')\,\d \mu(x)\d t\d \mu(x')\d t'\right|\\
&\le C(b-a)^{\alpha}\|g\|_{L^2_x(\d\mu)L^1_t}\|h\|_{L^2_x(\d\mu)L^1_t}.\nonumber
\end{align}
Moreover, for $0<\rho<\alpha$ there exists a constant $C$ such that
\begin{align}\label{ineq:HLS-type}
&\left|\iint\iint g(x,t)h(x',t')|x-x'|^{-\rho}\,\d \mu(x)\d t\d \mu(x')\d t'\right|\\
&\le C\|g\|_{L^2_x(\d\mu)L^1_t}\|h\|_{L^2_x(\d\mu)L^1_t}.\nonumber
\end{align}
Here, the both integrals are taken over $(x,t)$, $(x',t')\in I\times I$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:Young and HLS-type}]
Denoting $G(x)=\|g(x,\cdot)\|_{L^1}$ and $H(x')=\|h(x',\cdot)\|_{L^1}$,
\begin{align*}
&\left|\iint\iint g(x,t)h(x',t')\chi_{[a,b]}(x-x')\,\d\mu(x)\d t\d\mu(x')\d t'\right|\\
&\le\int_{-1}^1\int_{-1}^1G(x)H(x')\chi_{[a,b]}(x-x')\d\mu(x)\d\mu(x').
\end{align*}
By invoking the Cauchy--Schwarz inequality on $L^2(I\times I,\d\mu\d\mu)$ and \eqref{alpha-dimensional},
\begin{align*}
&\int_{-1}^1\int_{-1}^1G(x)H(x')\chi_{[a,b]}(x-x')\d\mu(x)\d\mu(x')\\
&\lesssim
\left(\iint G(x)^2\chi_{[a,b]}(x-x')\,\d\mu(x)\d\mu(x')\right)^\frac12
\left(\iint H(x')^2\chi_{[a,b]}(x-x')\,\d\mu(x)\d\mu(x')\right)^\frac12\\
&
\lesssim(b-a)^{\alpha}\|G\|_{L^2_x(\d\mu)}\|H\|_{L^2_x(\d\mu)}.
\end{align*}

Now, \eqref{ineq:HLS-type} follows from \eqref{ineq:Young special case}, immediately. In fact, by applying a dyadic decomposition,
\begin{align*}
&\left|\iint\iint g(x,t)h(x',t')|x-x'|^{-\rho}\,\d \mu(x)\d t\d \mu(x')\d t'\right|\\
&\lesssim\sum_{j=0}^\infty2^{\rho j}\iint G(x)H(x')\chi_{[2^{-j},2^{-j+1}]}(x-x')\,\d\mu(x)\d\mu(x')\\
&\lesssim\sum_{j=0}^\infty 2^{(\rho-\alpha)j}\|G\|_{L^2_x(\d\mu)}\|H\|_{L^2_x(\d\mu)}\\
&\lesssim\|G\|_{L^2_x(\d\mu)}\|H\|_{L^2_x(\d\mu)}
\end{align*}
whenever $\rho-\alpha<0$.
\end{proof}

\section{Some reduction arguments}\label{sec:reduction arguments}

\subsection{Proof of (Theorem \ref{thm:maximal estimate with mu} $\Longrightarrow$ Theorem \ref{pointwise convergence with mu})}

Fix an arbitrary $f\in H^s(\mathbb R)$. Then, it is enough to show that $\mu(\mathfrak D(\gamma,f))=0$. Now, choose a sequence $\{f_n\}_{n\in\mathbb N}\subset C_0^\infty(\mathbb R)$ which converges in $H^s$-norm to $f\in H^s(\mathbb R)$. Then, we divide the divergence set into localized pieces as follows and show that all terms turn out to be $0$.
\begin{align}\label{mu div-set is less than sum sum}
\mu(\mathfrak D(\gamma,f))
\le
\sum_{j\in\mathbb Z}\sum_{\ell=1}^\infty\mu(\{x\in I+j\,:\,\lim_{t\to0}|S_tf(\gamma(x,t))-f(x)|>\ell^{-1}\}).\nonumber
\end{align}
Now, for each $n \ge 1$, $j=0$ and $\lambda\ge1$ observe that
\begin{align*}
&\mu(\{x\in I\, :\, \lim_{t\to0}|S_tf(\gamma(x,t))-f(x)|>\lambda^{-1}\})\\
&\le
\mu(\{x\in I\, :\, \limsup_{t\to0}|S_tf(\gamma(x,t))-S_tf_n(\gamma(x,t))|>(3\lambda)^{-1}\})\\
&\qquad+
\mu(\{x\in I\, :\, \limsup_{t\to0}|S_tf_n(\gamma(x,t))-f_n(x)|>(3\lambda)^{-1}\})\\
&\qquad\quad+
\mu(\{x\in I\, :\, |f_n(x)-f(x)|>(3\lambda)^{-1}\})\\
&\le
\mu(\{x\in I\,:\,\sup_{t\in I}|S_t\big(f(\gamma(x,t))-f_n(\gamma(x,t))\big)|>(3\lambda)^{-1}\})\\
&\qquad+
0
+
\mu(\{x\in I\, :\, |f_n(x)-f(x)|>(3\lambda)^{-1}\}).
\end{align*}
By invoking the Chebyshev's inequality and Theorem \ref{thm:maximal estimate with mu} we obtain
\begin{equation}\label{mu div-set is less than f-f H2}
\mu(\{x\in I\, :\, \lim_{t\to0}|S_tf(\gamma(x,t))-f(x)|>\lambda^{-1}\})\lesssim\lambda^2\|f-f_n\|_{H^s(\mathbb R)}^2,
\end{equation}
which tends to $0$ as $n\to\infty$. For other $j$, make translation $x\mapsto x+j$ and we define a measure $\mu_j$ by $\mu_j(x) = \mu(x+j)$ and a curve $\gamma_j$ by $\gamma_j(x,t) = \gamma(x+j,t)$, both of which satisfy the required conditions\footnote{
Strictly speaking, $\gamma_j \not\in \Gamma(\kappa)$ because $\gamma_j(x,0)=x+j$, however this translation effect is negligible for Theorem \ref{thm:maximal estimate with mu} since $|\gamma_j(x,t)-\gamma_j(x',t')|$ is essentially equivalent to $|\gamma(x,t)-\gamma(x',t')|$ for any $x,x',t,t'$.

}
for Theorem \ref{thm:maximal estimate with mu} so that \eqref{mu div-set is less than f-f H2} holds with $I$ replaced by $I+j$. Therefore, for all $j\in\mathbb Z$ and $\ell\ge1$
\[
\mu(\{x\in I+j\,:\, \lim_{t\to0}|S_tf(\gamma(x,t))-f(x)|>\ell^{-1}\})=0
\]
holds as desired. \qed

\subsection{Proof of (Theorem \ref{pointwise convergence with mu} $\Longrightarrow$ Corollary \ref{divergence set estimate})}
Let $s>\frac14$ and $f\in H^s(\mathbb R)$. If we suppose $\dim_c(\mathfrak{D}(f,\gamma))>\max\{1-2s,\frac{1-2s}{m\kappa}\}\ge0$, then one would find $0<\alpha<1$ satisfying $\dim_c(\mathfrak{D}(f,\gamma))>\alpha>\max\{1-2s,\frac{1-2s}{m\kappa}\}\ge0$. Here, note that the second inequality is equivalent to $s>\max\{\frac{1-\alpha}{2},\frac{1-m\alpha\kappa}{2}\}$. Hence, by Lemma \ref{lem:Frostman} there would exist an $\alpha$-dimensional measure $\mu$ such that $\mu(\mathfrak{D}(f,\gamma))>0$, which contradicts Theorem \ref{pointwise convergence with mu}, and we must have $\dim_c(\mathfrak{D}(f,\gamma))\le\max\{1-2s,\frac{1-2s}{m\kappa}\}$. \qed

\section{Proof of Theorem \ref{thm:maximal estimate with mu}}\label{sec:proof of maximal estimate}

Let
\[
s_*=\min\left\{\frac14,\frac\alpha2,\frac{m\alpha\kappa}{2}\right\}.
\]
By following the standard steps via Littlewood--Paley decomposition, it is enough to show the following proposition. (For the details, for instance, see  \cite{Sh19}.)
\begin{proposition}\label{prop:Sf is less than lambda f with mu}
Let $\varepsilon>0$. Then, there exists a constant $C_\varepsilon$ such that
\begin{equation}\label{ineq:SPf is less than f with mu}
\left\|\sup_{t\in I}|S_tf(\gamma(\cdot,t))|\right\|_{L^2(I,\d\mu)}\le C_\varepsilon \lambda^{\frac12-s_*+\varepsilon}\|f\|_{L^2}
\end{equation}
holds for all $\lambda\ge1$ and $f\in L^2(\mathbb{R})$ whose Fourier support is contained in $\{\xi\in\mathbb{R}:\frac{\lambda}{2}\le|\xi|\le2\lambda\}$.
\end{proposition}
\begin{proof}[Proof of Proposition \ref{prop:Sf is less than lambda f with mu}]

Let
\[
Tf(x,t)=\chi(x,t)\int_{\mathbb{R}}e^{i(\gamma(x,t)\xi+t|\xi|^m)}f(\xi)\psi(\tfrac\xi\lambda)\,\d\xi,
\]
where $\chi=\chi_{I\times I}$ and $\psi\in C_0^\infty((-2,-\frac12)\cup(\frac12,2))$. Then, by Plancherel's theorem, \eqref{ineq:SPf is less than f with mu} follows from
\begin{equation}\label{ineq:Tf is less than f}
\|Tf\|_{L^2_x(\d\mu)L^\infty_t}^2\lesssim\lambda^{1-2s_*+\varepsilon}\|f\|_{L^2}^2.
\end{equation}
By duality, \eqref{ineq:Tf is less than f} is equivalent to
\begin{equation}\label{ineq:TF is less than F}
\|T^*F\|_{L^2}^2\lesssim\lambda^{1-2s_*+\varepsilon}\|F\|_{L^2_x(\d\mu)L^1_t}^2,
\end{equation}
where
\[
T^*F(\xi)=\psi(\tfrac{\xi}{\lambda})\iint \chi(x',t')e^{-i(\gamma(x',t')\xi+t'|\xi|^m)}F(x',t')\,\d\mu(x')\d t'.
\]
Then,
\begin{align*}
\|T^*F\|_{L^2}^2&=\int\psi(\tfrac{\xi}{\lambda})^2\iint\iint\chi(x,t)\chi(x',t')\\
&\qquad\times e^{i((\gamma(x,t)-\gamma(x',t'))\xi+(t-t')|\xi|^m)}\overline{F}(x,t)F(x',t')\,\d\mu(x)\d t\d\mu(x')\d t'\d\xi\\
&=\int_W\int_{W'}\chi(w)\chi(w')\overline{F}(w)F(w')K_\lambda(w,w')\,\d_\mu w\d_\mu w'\\
&=\sum_{\ell=1}^3\iint_{V_\ell}\chi(w)\chi(w')\overline{F}(w)F(w')K_\lambda(w,w')\,\d_\mu w\d_\mu w'\\
&=\mathcal I_1+\mathcal I_2+\mathcal I_3.
\end{align*}
Here, we denote $W=I\times I$, $w=(x,t)\in W$, $w'\in(x',t')\in W$ and $\d_\mu w=\d\mu(x)\d t$. Also,
\begin{align*}
K_\lambda(w,w')&=\int_\mathbb{R} e^{i\phi(\xi,w,w')}\psi(\tfrac{\xi}{\lambda})^2\,\d\xi\\
&=\lambda\int_\mathbb{R} e^{i\phi(\lambda\xi,w,w')}\psi(\xi)^2\,\d\xi,
\end{align*}
\[
\phi(\xi,w,w')=(\gamma(x,t)-\gamma(x',t'))\xi+(t-t')|\xi|^m
\]
and
\[
\begin{cases}
V_1=\{(w,w')\in W\times W:|x-x'|\le2\lambda^{-\frac{2s_*}{\alpha}}\},\\
V_2=\{(w,w')\in W\times W:|x-x'|>2\lambda^{-\frac{2s_*}{\alpha}}\ {\rm and}\ \frac1{C_2}|x-x'|> 2C_1|t-t'|^\kappa\},\\
V_3=\{(w,w')\in W\times W: |x-x'|>2\lambda^{-\frac{2s_*}{\alpha}}\ {\rm and}\  \frac1{C_2}|x-x'|\le2C_1|t-t'|^\kappa\}.
\end{cases}
\]
Then, \eqref{ineq:TF is less than F} follows from
\[
\mathcal I_\ell\lesssim\lambda^{1-2s_*+\varepsilon}\|F\|_{L^2_x(\d\mu)L^1_t}^2
\]
for each $\ell=1,2,3$.

\subsection*{The term $\mathcal I_1$}

By using the trivial estimate
\begin{equation}\label{ineq:trivial estimate}
|K_\lambda(w,w')|\lesssim\lambda
\end{equation}
and Lemma \ref{lem:Young and HLS-type}, we obtain
\begin{align*}
\mathcal I_1\lesssim\lambda^{1-2s_*}\|F\|_{L^2_x(\d\mu)L^1_t}^2.
\end{align*}

\subsection*{The term $\mathcal I_2$}

In this case, first note that we have
\begin{equation}\label{ineq:gamma-gamma is less than x-x}
|\gamma(w)-\gamma(w')|\ge\frac1{2C_2}|x-x'|
\end{equation}
for $(w,w')\in V_2$ by using \eqref{cond:Holder} and \eqref{cond:bilipschitz}. Next, we split $K_\lambda$ into $\mathcal K_1$ and $\mathcal K_2$ as follows.
\begin{align*}
K_\lambda(w,w')&=\lambda\int_{U_1}e^{i\phi(\lambda\xi,w,w')}\psi(\xi)^2\,\d\xi+\lambda\int_{U_2}e^{i\phi(\lambda\xi,w,w')}\psi(\xi)^2\,\d\xi\\
&=:\mathcal K_1+\mathcal K_2,
\end{align*}
where
\[
\begin{cases}
U_1=\{\xi\in\supp\psi:\frac1{C_2}|x-x'|>4m\lambda^{m-1}|t-t'||\xi|^{m-1}\},\\
U_2=\{\xi\in\supp\psi:\frac1{C_2}|x-x'|\le 4m\lambda^{m-1}|t-t'||\xi|^{m-1}\}.
\end{cases}
\]
For $\mathcal K_1$,
we use \eqref{ineq:gamma-gamma is less than x-x} in order to estimate the phase
\begin{align*}
\left|\frac{\d}{\d\xi}\phi(\lambda\xi,w,w')\right|&\ge\lambda|\gamma(w)-\gamma(w')|-m\lambda^m|t-t'||\xi|^{m-1}\\
&\ge\frac1{2C_2}\lambda|x-x'|-m\lambda^m|t-t'||\xi|^{m-1}\\
&>\frac1{4C_2}\lambda|x-x'|\\
&\gtrsim\lambda^{1-\frac{2s_*}{\alpha}}\\
&>1
\end{align*}
since $\frac{2s_*}{\alpha}=\min\{\frac{1}{2\alpha},1,m\kappa\}\le1$. Here, note that the interval $U_1$ consists of at most two intervals since $\frac{\d}{\d\xi}\phi(\lambda\xi,w,w')$ is monotone on each interval $(-\infty,-1]$ and $[1,\infty)$. Thus, Lemma \ref{lem:van der Corput} gives that
\begin{align*}
\mathcal K_1\lesssim\lambda(\lambda|x-x'|)^{-1}\lesssim\lambda(\lambda|x-x'|)^{-\min\{\frac12,\alpha\}}.
\end{align*}

On the other hand, for $\mathcal K_2$,
\begin{align*}
\left|\frac{\d^2}{\d\xi^2}\phi(\lambda\xi,w,w')\right|&\gtrsim\lambda^m|t-t'||\xi|^{m-2}\\
&\gtrsim\lambda|x-x'|
\end{align*}
so that we are allowed to apply Lemma \ref{lem:van der Corput} to obtain
\begin{align*}
\mathcal K_2&\lesssim\lambda(\lambda|x-x'|)^{-\frac12}\\
&\lesssim\lambda(\lambda|x-x'|)^{-\min\{\frac12,\alpha\}}.
\end{align*}
Hence, for $(w,w')\in V_2$ we have the following kernel estimate
\begin{align*}
|K_\lambda(w,w')|&\lesssim\lambda^{1-\min\{\frac12,\alpha\}}|x-x'|^{-\min\{\frac12,\alpha\}}\\
&\lesssim\lambda^{1-\min\{\frac12,\alpha\}+\varepsilon}|x-x'|^{-\min\{\frac12,\alpha\}+\varepsilon}.
\end{align*}
Here, we used the separation assumption, $|x-x'|\gtrsim\lambda^{-\frac{2s_*}{\alpha}}$.
By Lemma \ref{lem:Young and HLS-type} with $\rho=\min\{\frac12,\alpha\}-\varepsilon<\alpha$ we conclude that
\begin{align*}
\mathcal I_2&\lesssim\lambda^{1-\min\{\frac12,\alpha\}+\varepsilon}\|F\|_{L^2_x(\d\mu)L^1_t}^2\\
&\lesssim\lambda^{1-2s_*+\varepsilon}\|F\|_{L^2_x(\d\mu)L^1_t}^2.
\end{align*}

\subsection*{The term $\mathcal I_3$}

 It remains to consider $\mathcal I_3$. Observe that
\begin{align*}
\left|\frac{\d^2}{\d\xi^2}\phi(\lambda\xi,w,w')\right|&\gtrsim\lambda^m|t-t'||\xi|^{m-2}\\
&\gtrsim\lambda^m|x-x'|^\frac1\kappa\\
&\gtrsim\lambda^m\lambda^{-\frac{2s_*}{\alpha\kappa}}\\
&>1
\end{align*}
since $\frac{2s_*}{\alpha\kappa}=\min\{\frac{1}{2\alpha\kappa},\frac{1}{\kappa},m\}\le m$. Then, by Lemma \ref{lem:van der Corput} for arbitrary small $\varepsilon>0$
\begin{align*}
|K_\lambda(w,w')|&\lesssim\lambda(\lambda^m|x-x'|^\frac1\kappa)^{-\frac12}\\
&\lesssim\lambda(\lambda^m|x-x'|^{\frac1\kappa})^{-\frac{2s_*}{m}}\\
&\sim\lambda^{1-2s_*}|x-x'|^{-\frac{2s_*}{m\kappa}}\\
&\lesssim\lambda^{1-2s_*+\varepsilon}|x-x'|^{-\frac{2s_*}{m\kappa}+\varepsilon}
\end{align*}
since $\frac{2s_*}{m}=\min\{\frac{1}{2m},\frac{\alpha}{m},\alpha\kappa\}<\frac12$ and our separation assumption again. Therefore, applying Lemma \ref{lem:Young and HLS-type} with $\rho= \frac{2s_*}{m\kappa}-\varepsilon=\min\{\frac{1}{2m\kappa},\frac{\alpha}{m\kappa},\alpha\}-\varepsilon<\alpha$, it follows that
\[
\mathcal I_3\lesssim\lambda^{1-2s_*+\varepsilon}\|F\|_{L^2_x(\d\mu)L^1_t}^2.
\]

\end{proof}

\section{The necessary conditions regarding Theorem \ref{thm:maximal estimate with mu}}\label{sec:counterexamples}

In this section, we present $s\ge\max\left\{\frac14, \frac{1-\alpha}{2}, \frac{1-m\alpha\kappa}{2} \right\}$ is necessary for Theorem \ref{thm:maximal estimate with mu}, otherwise there exist $\gamma\in\Gamma(\kappa)$ and $\alpha$-dimensional measure $\mu$ such that \eqref{ineq:maximal estimate with mu} fails.
Throughout the section, we shall let $\lambda\ge1$, $\gamma(x,t)=x-t^\kappa$, $\mu(x)=|x|^{-1+\alpha}\,\d x$ and $\psi_0$ be a smooth radial bump function whose support is in a small neighborhood of the origin. Also, we fix $m>1$ and $0<\kappa\le1$, and we assume that the maximal estimate \eqref{ineq:maximal estimate with mu} holds.

\subsection*{The necessity of $s\ge\max\{\frac{1-\alpha}{2},\frac{1-m\alpha\kappa}{2}\}$}

In this case, we will simply follow the idea in \cite{CLV12}.
Let
\[
\widehat{f_1}(\xi)=\psi_0(\lambda^{-\frac1m}\xi).
\]
With this initial data,
\begin{align*}
|S_tf_1(\gamma(x,t))|&\sim\left|\int e^{i((x-t^\kappa)\xi+t|\xi|^m)}\widehat{f_1}(\xi)\,\d\xi\right|\\
&=\lambda^{\frac1m}\left|\int e^{i\phi_1(\eta,x,t)}\psi_0(\eta)\,\d\eta\right|,
\end{align*}
where \[
\phi_1(\eta,x,t)=\lambda^{\frac1m}(x-t^\kappa)\eta+\lambda t|\eta|^m.
\]
For
$x\in(0,\frac{1}{100}(\lambda^{-\frac1m}+\lambda^{-\kappa}))$ and $|t|<\frac{1}{100}\lambda^{-1}$, we have
\[
|\phi_1(\eta,x,t)|=|\lambda^{\frac1m}(x-t^\kappa)\eta+\lambda t|\eta|^m)|\le\frac12
\]
so that
\begin{align*}
|S_tf_1(\gamma(x,t))|
&\gtrsim
\lambda^{\frac1m}\left|\int(\cos\phi_1(\eta,x,t))\psi_0(\eta)\,\d\eta\right|\\
&\gtrsim
\lambda^\frac1m\chi_{(0,\frac{1}{100}(\lambda^{-\frac1m}+\lambda^{-\kappa}))\times(0,\frac{1}{100}\lambda^{-1})}(x,t).
\end{align*}
Hence,
\begin{align*}
\left\|\sup_{t\in I}|S_tf_1(\gamma(\cdot,t))|\right\|_{L^2(I,\d\mu)}
&\ge
\left\|\sup_{t\in(0,\frac{1}{100}\lambda^{-1})}|S_tf(\gamma(\cdot,t))|\right\|_{L^2((0,\frac{1}{100}(\lambda^{-\frac1m}+\lambda^{-\kappa})),\d\mu)}\\
&\gtrsim\lambda^\frac1m\max\{\lambda^{-\frac\alpha m},\lambda^{-\alpha\kappa}\}^{\frac12}.
\end{align*}
On the other hand,
\begin{align*}
\|f_1\|_{H^s}&\sim\left(\int(1+|\xi|^2)^s|\psi_0(\lambda^{-\frac1m}\xi)|^2\,\d\xi\right)^\frac12\\
&\lesssim\lambda^{\frac sm}\lambda^{\frac{1}{2m}}.
\end{align*}
Therefore, combining the above calculations, we obtain
\[
\lambda^\frac1m\max\{\lambda^{-\frac\alpha m},\lambda^{-\alpha\kappa}\}^{\frac12}\lesssim\lambda^{\frac sm}\lambda^{\frac{1}{2m}}.
\]

If $\frac1m\ge\kappa$, then $\max\{\lambda^{-\frac\alpha m},\lambda^{-\alpha\kappa}\}=\lambda^{-\alpha\kappa}$, and as $\lambda\to\infty$ it is necessary that
\[
\frac 1m-\frac{\alpha\kappa}{2}\le\frac sm+\frac{1}{2m},
\]
which clearly gives
\[
s\ge\frac{1-m\alpha\kappa}{2}.
\]

If $\frac1m<\kappa$, then $\max\{\lambda^{-\frac\alpha m},\lambda^{-\alpha\kappa}\}=\lambda^{-\frac\alpha m}$, and as $\lambda\to\infty$, it is necessary that
\[
\frac1m-\frac{\alpha}{2m}\le\frac sm+\frac{1}{2m},
\]
which is
\[
s\ge\frac{1-\alpha}{2}.
\]

\subsection*{The necessity of $s\ge\frac14$}

In this case, we will refer to the idea in \cite{Sj87} (see page 712). Let
\[
\widehat{f_2}(\xi)=\lambda^{-1}\psi_0(\lambda^{-1}\xi+\lambda).
\]
Then, by the change of variables $-\eta=\lambda^{-1}\xi+\lambda$,
\begin{align*}
|S_tf_2(\gamma(x,t))|&\sim\left|\int e^{i((x-t^\kappa)\xi+t|\xi|^m)}\lambda^{-1}\psi_0(\lambda^{-1}\xi+\lambda)\,\d\xi \right|\\
&=\left|\int e^{i\phi_2(\eta,x,t)}\psi_0(-\eta)\,\d\eta\right|,
\end{align*}
where
\[
\phi_2(\eta,x,t)=-(x-t^\kappa)\lambda\eta+\lambda^m t|\lambda+\eta|^m.
\]
By a Taylor expansion,
\begin{align*}
(\lambda+\eta)^m&=\lambda^m(1+\lambda^{-1}\eta)^m\\
&=\lambda^m\left(1+m\lambda^{-1}\eta+\frac{m(m-1)}{2}\lambda^{-2}\eta^2+O(\lambda^{-3}|\eta|^3)\right)\\
&=\lambda^m+m\lambda^{-(1-m)}\eta+\frac{m(m-1)}{2}\lambda^{-(2-m)}\eta^2+O(\lambda^{-(3-m)}|\eta|^3),
\end{align*}
and it follows that
\begin{align*}
\phi_2(\eta,x,t)
&=
-\lambda x\eta+\lambda t^\kappa\eta+\lambda^{2m}t+m\lambda^{-(1-2m)}t\eta\\
&\qquad+
\frac{m(m-1)}{2}\lambda^{-(2-2m)}t\eta^2+O(\lambda^{-(3-2m)}t|\eta|^3)\\
&=
\lambda^{2m}t+\lambda (-x+t^\kappa+m\lambda^{-(2-2m)}t)\eta\\
&\qquad+
\frac{m(m-1)}{2}\lambda^{-(2-2m)}t\eta^2+O(\lambda^{-(3-2m)}t|\eta|^3).
\end{align*}
For $x\in(0,\frac{1}{100})$, we can choose $t(x)$ such that $x=t(x)^\kappa+m\lambda^{-(2-2m)}t(x)$. In fact, if we consider the function $\tau(t) = t^{\kappa} +m\lambda^{-(2-2m)}t$, then $\tau:[0,\infty)\to[0,\infty)$ is a strictly increasing bijection and
$$0=\tau^{-1}(0)<\tau^{-1}(x)=t(x)<\tau^{-1}(\tfrac{1}{100})<\frac{\lambda^{2-2m}}{100m}.$$
Therefore, for such choice of $(x,t(x))$, it follows that
\[
|\phi_2(\eta,x,t(x))-\lambda^{2m}t(x)|\lesssim 0+\frac{1}{100}+O(\lambda^{-1})\le\frac12,
\]
which implies that
\begin{align*}
|S_tf_2(\gamma(x,t(x)))|
&\sim
\left|\int\cos(\phi_2(\eta,x,t(x))-\lambda^{2m}t(x))\psi_0(-\eta)\,\d\eta\right|\\
&\gtrsim
\chi_{(0,\frac{1}{100})}(x).
\end{align*}
Hence,
\[
\left\|\sup_{t\in I}|S_tf_2(\gamma(\cdot,t))|\right\|_{L^2(I,\d\mu)}
\gtrsim
1.
\]
On the other hand,
\begin{align*}
\|f_2\|_{H^s}&=\left(\int(1+|\xi|^2)^s|\lambda^{-1}\psi_0(\lambda^{-1}\xi+\lambda)|^2\,\d\xi\right)^\frac12\\
&\lesssim\lambda^{2s}\lambda^{-\frac12}.
\end{align*}
Therefore, combining the calculations above implies that
\[
1\lesssim \lambda^{2s}\lambda^{-\frac12}.
\]
As $\lambda\to\infty$, it is necessary that
\[
s\ge\frac14.
\]
This ends the proof that $s\ge\max\left\{\frac14, \frac{1-\alpha}{2}, \frac{1-m\alpha\kappa}{2} \right\}$ is necessary for \eqref{ineq:maximal estimate with mu} to hold.

\subsection*{Acknowledgment} The authors would like to thank (the second author's adviser) Neal Bez for the constant encouragement and constructive suggestions. The second author also wishes to express his great appreciation to Sanghyuk Lee for introducing him to the problem.

config: arxiv


TITLE: How do you integrate $\int x^3\operatorname{sech}^2(x^2)\,\mathrm{d}x\:?$
QUESTION [0 upvotes]: How do you integrate $$\int x^3\operatorname{sech}^2(x^2)\,\mathrm{d}x\:?$$

I've tried integration by parts, but then I find myself left with having to integrate $\operatorname{sech}(x^2)^2$ if i let it equal to $dv/dx$ (and I don't know how to work it out if i let $dv = x^3$ instead).
And I don't think you can use derivative substitution considering you can't write $x^3$ in terms of the derivative of $x^2$. So how would you solve it?

REPLY [1 votes]: If you do $x^2=u$ and $2x\,\mathrm dx=\mathrm dy$, then your integral becomes$$\frac12\int y\operatorname{sech}^2(y)\,\mathrm dy.$$Now, use integration by parts:\begin{align}\int y\operatorname{sech}^2(y)\,\mathrm dy&=y\tanh(y)-\int\tanh(y)\,\mathrm dy\\&=y\tanh(y)-\log\bigl(\cosh(y)\bigr)\\&=x^2\tanh\left(x^2\right)-\log\left(\cosh\left(x^2\right)\right).\end{align}