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Generalized family of Hölder inequalities

Is the "only if" direction of the following fact known? For fixed sequences $(a)_i = a_1, \dots, a_r$, $(b)_i = b_1, \dots, b_r$ and $(c)_i = c_1, \dots, c_r$, the inequality $\prod_{i = 1}^...
Anonymous's user avatar
3 votes
0 answers
217 views

Is this integral operator about Stokes' Flow compact?

Consider the following integral operator $\mathcal{A}$ on [EDITED: continuous vector function $f=(f_i):\partial S\to{\mathbb R}^3$]: $$ ({\mathcal A}f)_j(x_0):=\int_{\partial S}\sum_{i=1}^3 f_i(x)G_{...
user avatar
3 votes
0 answers
251 views

What is the origin of the metrization problem for compact convex sets?

The following is an ``old question in analysis:'' Is it true that every perfectly normal compact convex subset of a locally convex topological vector space is metrizable? Here perfectly normal means ...
Justin Moore's user avatar
  • 3,547
3 votes
0 answers
223 views

Extension of positive operators and Bauer-Namioka

When $X$ is a vector subspace of an ordered vector space $A$, any positive linear functional $f: X \to R$ extends to all of $A$ as a positive linear functional provided one can find a nonvoid, ...
Dieter K's user avatar
2 votes
0 answers
228 views

Any rigorous construction of $\phi^4$ theories without the mass term in the Lagrangian? (revised)

There are various papers on rigorous construction of massive $\phi^4$ theories in $2$ or $3$ Euclidean dimensions. In 2D, there are in fact more general results such as this one by Glimm, Jaffe and ...
Isaac's user avatar
  • 3,477
2 votes
0 answers
54 views

Distance between a Hölder function and a Sobolev ball

Let $\Omega$ denote $[0, 1]^n$ and let $\|\cdot\|_{k, p}$ and $|\cdot|_{m, \alpha}$ denote norms of Sobolev space $W^{k,p}(\Omega)$ and Holder space $C^{m, \alpha}(\Omega)$, respectively. My question ...
Drew Brady's user avatar
2 votes
0 answers
82 views

What is Lipschitz constant of the radial renormalization $(X,\|\cdot\|_a) \rightarrow (X,\|\cdot\|_b)$ on a normed vector space $X$

Suppose that $X$ is a vector space with two norms $\|\cdot\|_a$ and $\|\cdot\|_b$. The mapping $$ f(x) = \frac{\|x\|_{a}}{\|x\|_{b}} x, \qquad \forall x \in X, $$ with $f(0)=0$ is a radial and maps ...
shuhalo's user avatar
  • 5,327
2 votes
0 answers
74 views

References for a class of Banach space-valued Gaussian processes

Let $E$ be a separable Banach space, consider a centered $E$-valued Gaussian process $\{x_t,t\ge 0\}$ that satisfies \begin{equation} \mathbb{E}\phi(x_s)\psi(x_t)=R(s,t)K(\phi,\psi),\quad \phi,\psi\in ...
Jorkug's user avatar
  • 121
2 votes
0 answers
138 views

Is $L^2([a,b]; L^2(S^2))$ the same as $L^2([a,b] \times S^2)$?

The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm $$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$ The space $L^2([a,b]\times S^2)$ ...
Laithy's user avatar
  • 969
2 votes
0 answers
111 views

Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$

Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it. We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb ...
Akira's user avatar
  • 835
2 votes
0 answers
89 views

On dense subspaces of $L^p$-spaces of finitely additive measures

Let $\mu$ be a finite, finitely additive measure defined on the Borel $\sigma$-algebra of a real separable Hilbert space $\mathcal{H}$ with dual $\mathcal{H}^{*}$. Write $L^{p}(\mathcal{H},\mu)$ for ...
S.Z.'s user avatar
  • 505
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0 answers
95 views

Self adjoint operators from energy functionals

It is known that the equation $$ \Delta f = 0 $$ on some bounded domain $\Omega$ on $\mathbb{R}^n$ subjected to certain boundary conditions can be derived through the minimization of the Dirichlet ...
user8469759's user avatar
2 votes
0 answers
62 views

Continuous-time Wold decomposition

I'm looking for a reference for the Wold–Zasukhin decomposition in continuous time for stationary random processes on the real line. I am aware of the classic result in the book from Rozanov, which ...
arknas's user avatar
  • 21
2 votes
0 answers
124 views

Uniqueness in interpolation of Hilbert spaces

I am wondering under what condition it is true that for Hilbert spaces $H$ and $H_0$ such that $H \hookrightarrow H_0$ there is uniqueness in the existence of a Hilbert space $H_1 \hookrightarrow H$ ...
rafoub92's user avatar
2 votes
0 answers
126 views

Differential equations: trying to connect a nonlinear equation to a linear one

The following is motivated by taking a product space $\Omega$ and splitting it into two parts via projections, whose subspaces, $T$ and $X$, are home to functions which satisfy a nonlinear PDE and a ...
John McManus's user avatar
2 votes
0 answers
325 views

Examples of RKHS that are "classical"

Among the so-called "classical" Hilbert spaces ($L^2$, Sobolev, Hardy, Bergman, etc.), which are very well-studied, which are RKHSs? It is easy to construct example of RKHSs by applying the ...
lost_analyst's user avatar
2 votes
0 answers
83 views

Singular integral operators acting on Zygmund class

It is proven in "Classical and Modern Fourier Analysis" by L. Grafakos (Corollary 6.7.2) that if a kernel $K(x)$ defined away from the origin on $\mathbb{R}^n$ satisfies $$\sup_{0<R<\...
MMagana's user avatar
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2 votes
0 answers
62 views

On a real smooth version of white noise distribution theory

In white noise analysis, one starts with a real Gelfand triple $\mathcal{N}\subset \mathcal{H} \subset \mathcal{N}^{*}$ and produces out of it, using complexifications along the way, the complex ...
S.Z.'s user avatar
  • 505
2 votes
0 answers
104 views

Schwarz space on the upper half-plane

Let us think of the Schwartz space $\mathcal{S}(\mathbb{R}^2_+)$ on the upper half-plane $\mathbb{R}^2_+=\mathbb{R}\times(0,+\infty)$ defined as $$ \mathcal{S}(\mathbb{R}^2_+)=\left\{f\in C^\infty(\...
Bedovlat's user avatar
  • 1,959
2 votes
0 answers
64 views

Scaling limit of ODE with double-well potential

Let us consider the ODE $$ \frac{d}{dt}x_\epsilon(t) = -\frac{1}{\epsilon} W'(x_\epsilon(t)) $$ where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the ODE reads $$...
Riku's user avatar
  • 839
2 votes
0 answers
155 views

Second differential of total variation

I am trying to give meaning to the notion of second differential of total variation. For sufficiently regular $u:\Omega \subset \mathbb{R}^2 \to \mathbb{R}$ let the total variation be given by $$TV(u)=...
Marko Rajkovic's user avatar
2 votes
0 answers
149 views

Reference for weighted Sobolev spaces

I'm looking for a comprehensive reference illustrating, from the ground up, the basics of weighted Sobolev Spaces on Lipschitz domains (this case should be included, but I don't need less than it). ...
Lilla's user avatar
  • 235
2 votes
0 answers
161 views

The Laplace transform and the Lagrange compositional inversion formula

I'm looking for references which derive the Lagrange inversion formula, given below (in bold), for the Taylor series coefficients of the compositional inverse of a function $f$ analytic at the origin ...
Tom Copeland's user avatar
  • 10.5k
2 votes
0 answers
559 views

Multiplication in Sobolev space with negative exponent

My initial problem is the following: I would like to estimate $\lVert f^2\rVert_{H^{-2}}$ in the sense of Sobolev embeddings, where $f:\Omega \rightarrow \mathbb{R}$ is a function defined on a bounded ...
Paul's user avatar
  • 914
2 votes
0 answers
108 views

Characterization of inverse limits of finite-dimensional convex cones

Consider a countable inverse system $C_1\substack{f_1 \\ \leftarrow} C_2 \substack{f_2 \\ \leftarrow} C_3 \substack{f_3 \\ \leftarrow} \ldots$ where the $C_i$ are finite-dimensional convex cones of ...
postdoc's user avatar
  • 21
2 votes
0 answers
110 views

About Hilbert-Haar theory

The Hilbert-Haar theory says that functionals $\mathcal{F}(u,B)=\int_{B} F(\nabla u)\,dx$, where $F$ is a convex function and $B$ is a bounded domain in $\mathbb{R}^N$, take a minimum in the space of ...
Thangachelli Debopritama's user avatar
2 votes
0 answers
85 views

Functions with smooth projections on finite-dimensional subspaces

Let $E,F$ be Banach spaces and $F$ be finite-dimensional and $E$ be strictly convex. Let $f\in C(F,E)$ have the property that: $$ \text{For every finite-dimensional subspace $E'\subseteq E$ we have } ...
ABIM's user avatar
  • 5,405
2 votes
0 answers
125 views

Regularity up to boundary of a solution $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ to $\Delta^2 u = -\text{div}\, F$

Let $\Omega\subset \Bbb R^n$ be a $C^{2}$ domain (open and bounded) and let $p\in(1,\infty)$. Suppose $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ is a weak solution to the fourth-order elliptic system ...
BigbearZzz's user avatar
  • 1,245
2 votes
0 answers
65 views

Reference request for type of specific integral equation in two variable:

Consider the following integral equation: $$\int_0^\infty K(t,y)\phi(t,x)dt=0$$ Here, $K(t,y)$ is a trigonometric kernel and $\phi(t,x)$ is monotonic wrt $x$ ( for fixed $t$). I want to find the ...
GSA_1's user avatar
  • 41
2 votes
0 answers
654 views

Convergence of operator in norm resolvent sense and their eigenvectors

Let $\{T_n\}_{n=1}^\infty$ and $T$ be (unbounded) self-adjoint operators and $T_n\to T$ in norm resolvent sense, that is, for some $z\in \mathbb{C} \setminus \mathbb{R}$, $\|(zI- T_n)^{-1}- (zI- T)^{-...
user270619's user avatar
2 votes
0 answers
72 views

Product of Besov and Lorentz functions

Let us fix $n\in\mathbb{N}^+$ and $p,q\in [1,\infty)$. Given $r_1,r_2,r_3\in[1,\infty)$, I would like to understand whether we have the bound $$ \|fg\|_{L^{q,r_3}(\mathbb{R}^n)}\lesssim \|f\|_{B^{n/p}...
RaffaeleScandone's user avatar
2 votes
0 answers
145 views

Are Weyl sequences polynomially bounded?

Look at the Hilbert space $l^2( \mathbb{Z}) $ and let $A$ be a translation invariant band operator. I.e. if $\{ e_n \}_{n \in \mathbb Z} $ is the standard basis for $l^2( \mathbb{Z}) $ then it holds ...
Frederik Ravn Klausen's user avatar
2 votes
0 answers
169 views

Functions whose Fourier coefficients satisfy $ \sum_{k=1}^\infty |c_k| < 1 $?

Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to ...
Riku's user avatar
  • 839
2 votes
0 answers
86 views

Eigenvalues of the operator $A = -v'' + B(x) v$

How can I prove that for the eigenvalues of the operator $$A := -v'' + B(x) v$$ on $(0,L)$ with zero Dirichlet boundary condition it holds that $$ \left| \lambda_n - \frac{\pi^2n^2}{L^2}\right| \le ||...
Lao's user avatar
  • 217
2 votes
0 answers
134 views

Stone–Weierstrass theorem for stronger topologies

The Stone–Weierstrass theorem gives an easy to check criterion on a (algebra) set of functions $D\subseteq C(X)$ which ensures that $D$ is dense. Are any similar results for density in $C_b(X)$ ...
ABIM's user avatar
  • 5,405
2 votes
0 answers
181 views

Lyapounov's inequality for Orlicz norms

When a sequence $f \in \ell_1$, there is a very simple bound on its $\ell_q$-norms given by $\|f\|_q^q \leq \|f\|_1 \cdot \|f\|_\infty^{q-1}$. This inequality is a special (or rather limit) case of ...
ARG's user avatar
  • 4,432
2 votes
0 answers
53 views

References for linear relations on Hilbert spaces

I am trying to find a reference for linear relations (multivalued operators). I would like to have something which gives an introductory overview. All I have found so far doesn't seem right for ...
Nathanael Skrepek's user avatar
2 votes
0 answers
136 views

Reference of a modified version of Hahn-Banach Theorem

I need a reference on where to find a modified version of Hahn-Banach Theorem that ensures that if $F(g) = \int g d\mu$ defines a positive linear functional on a subspace of $C_{0}(M, \mathbb{R})$, $F$...
Ilovemath's user avatar
  • 677
2 votes
0 answers
62 views

Existence and uniqueness for semilinear problem

Consider the following problem: $$-\Delta u + [(u)^+]^\alpha = 0,$$ where $(\cdot)^+$ is the positive part function and $\alpha >0$. How does the theory of monotone operators provide existence ...
user avatar
2 votes
0 answers
154 views

Browder's Fixed Point Theorem in uniformly convex Banach spaces with non-identical image

This is in fact an exercise from Dirk Werner's book "Funktionalanalysis", but I do think that the result is quite interesting and up to now, I can only partly solve this problem. From the point of my ...
Peter's user avatar
  • 223
2 votes
0 answers
95 views

Exp-decay estimate of Schrodinger equation

Consider the equation $Hu=0$ with $u\in L^2(\Omega)$, where $H=-\Delta+V$ for some bounded continuous function $V$ and $\Omega$ is an un-bounded domain(e.g. $\mathbb R^n$). If $0$ is in discrete ...
DLIN's user avatar
  • 1,915
2 votes
0 answers
122 views

A new topology on the dual of a locally convex space?

Working with the separable quotient problem for locally convex spaces we (with Saak Gabriyelyan) arrived to an interesting topology on the dual of a locally convex space and we would like to know if ...
Taras Banakh's user avatar
  • 41.8k
2 votes
0 answers
199 views

Convergence of the difference quotient of a BV function

Consider a BV function $u \in BV(\mathbb{R}^N; \mathbb{R}^N)$. What can be said about the difference quotient $$ \frac{u(x+\epsilon y)-u(x)}{\epsilon} $$ regarding its convergence as $\epsilon \to 0$...
Riku's user avatar
  • 839
2 votes
0 answers
165 views

Jacobian and Jacobian matrix of solutions of ODE with Sobolev vector field

Let $\Phi$ be the Lagrangian flow (defined as in page 6 of this paper) of the ODE $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\ \Phi(x,0) = x \quad x \in \mathbb{R}^N \end{...
user avatar
2 votes
0 answers
279 views

Relationship between $p$-capacity and Riesz $s$-capacity of a set

What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set? Are they equivalent? If not, what inequalities hold? What is the difference (in terms ...
Riku's user avatar
  • 839
2 votes
0 answers
62 views

Differences among various index theories in critical point theory

Index theories help characterize critical points of functionals having certain symmetries. What are the differences (regarding problems they can be applied to) between for example these ones? the ...
Riku's user avatar
  • 839
2 votes
0 answers
255 views

Sobolev Multiplication on non-compact manifold

We know that for a compact Riemannian $n$-dim manifold $(M,g)$(the boundary could be nonempty), the Sobolev Multiplication Theorem states that $L^p_k\times L^q_l⟶L^r_m$, where $1/r−m/n>1/p−k/m+1/...
DLIN's user avatar
  • 1,915
2 votes
0 answers
164 views

Convergence of Eigenvalues and Eigenvectors for Uniformly Form-Bounded Operators

Suppose that $A$ is an operator on a dense domain $D(A)\subset L^2$ with compact resolvent, and with quadratic form $q(f,g):=\langle f,Ag\rangle$. Let $(r_n)_{n\in\mathbb N}$ be a sequence of ...
user78370's user avatar
  • 891
2 votes
0 answers
219 views

Integral with product of two infinite sums

I am looking for references and results on integrals with product of two infinite sums: $$S_1=\int_0^{\infty} \sum_{n=0}^{\infty} f(nx) \sum_{k=0}^{\infty} \overline{ g(kx)} dx$$ Above integral is ...
Bertrand's user avatar
  • 1,199
2 votes
0 answers
133 views

Convex hull of piece-wise linear functions

Let $K>1$ be a positive integer. Consider a function class $$\mathcal{F}_K:=\Big\{\max_{1\leq k\leq K} a_k^\top x + b_k:\ ||a_k||\leq 1, |b_k|\leq 1, \forall 1\leq k\leq K \Big\}$$ on some compact ...
O. Richard's user avatar

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