User Akira

1 vote

2 answers

188 views

Does a measurable $F :[0, T] \to L^p (\mathbb R^d; \mathbb R_{\ge 0})$ have a "flattened" measurable version?

asked Aug 9, 2023 at 20:52

1 vote

1 answer

118 views

Is there $\varepsilon \in (0, 1)$ such that $\sup_{t \in [0, \varepsilon]} [\ell_t]_\beta < \infty$?

asked Jun 1 at 15:04

1 vote

1 answer

86 views

Is it true that $\xi \in \partial G (v)$ implies $\frac{\xi}{F'(\phi (v))} \in \partial \phi (v)$?

asked Dec 14, 2023 at 15:22

1 vote

1 answer

277 views

Is it true that $F(X_0, \cdot) = X_0 + \int_0^T \sigma(s, X_0) \, \mathrm d B_s$ a.s.?

asked May 15 at 16:02

1 vote

1 answer

115 views

Is a $\sigma$-algebra generated by complete independent $\sigma$-algebras also complete?

asked May 19 at 8:36

1 vote

2 answers

130 views

Let $D$ be the set of those $\omega \in \Omega$ such that $f(\omega, \cdot)$ is $\mu$-integrable. Is $D$ measurable?

asked May 22 at 9:58

1 vote

1 answer

83 views

Lipschitz approximation of a probability measure with finite $1$-st moment by the ones with finite $p$-th moment

asked Mar 27 at 23:05

1 vote

2 answers

90 views

Is the difference between $\alpha$-Hölder constants of $f\rho$ and $g\rho$ controlled by $\|f-g\|_\infty$?

asked Apr 1 at 9:19

1 vote

1 answer

58 views

Lower bound the best $\alpha$-Hölder constant of a convolution

asked Apr 1 at 10:57

1 vote

0 answers

46 views

Reference request: Hölder regularity of $(1-\Delta)^{\frac{\alpha}{2}}$ for $\alpha >0$

asked Oct 25, 2023 at 16:24

1 vote

0 answers

83 views

For any $\beta>0$, there is a constant $c>0$ such that $\left\|(1-\Delta)^{\frac{\beta}{2}} f\right\|_{\infty} \leq c\|f\|_{C_b^\beta}$

asked Jan 9 at 17:50

1 vote

1 answer

106 views

Upper bound $I (t) := \sup_{x \in \mathbb R^d} \int_{\mathbb R^d} \frac{|x-y|^\alpha}{t^{d/2}} \exp ( - \frac{|\psi(x) - y|^2}{t} ) \, \mathrm d y$

asked Jan 17 at 11:04

1 vote

1 answer

113 views

An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y$

asked Jan 18 at 19:48

1 vote

1 answer

66 views

Upper bound $I_R := \int_{B_R^c} |x| (P_t \ell_\nu) (x) \, \mathrm d x$ in terms of $R, \nu, t$?

asked Jan 25 at 20:39

1 vote

1 answer

150 views

Is the Boltzmann entropy continuous in the supremum norm?

asked Nov 22, 2023 at 11:47

0 votes

1 answer

235 views

If we don't care about uniqueness, can we relax the coercivity condition in Lax-Milgram theorem?

asked Nov 25, 2023 at 21:33

0 votes

1 answer

185 views

Can we approximate a Hölder pdf by higher-order Hölder pdf's?

asked Nov 27, 2023 at 9:40

0 votes

1 answer

107 views

An identity about Bessel potential operators

asked Oct 25, 2023 at 16:39

0 votes

1 answer

54 views

How is this interpolating curve well-defined in the minimizing movement scheme?

asked Nov 10, 2023 at 17:05

0 votes

2 answers

262 views

Is the Boltzmann entropy lower semi-continuous in the weak topology induced by $C_b (\mathbb R^d)$?

asked Nov 2, 2023 at 12:32

0 votes

1 answer

88 views

Can we lower bound this entropy by $\int_{\mathbb R^d} \rho^k (x) \, \mathrm d x$ and $\int_{\mathbb R^d} |x|^2\rho (x) \, \mathrm d x$?

asked Nov 7, 2023 at 10:38

0 votes

0 answers

44 views

Are there probability densities $\rho, f_n$ such that $\lim_n \frac{[\rho * f_n]_\alpha}{\|\rho * f_n\|_\infty} = \infty$?

asked Apr 1 at 15:42

0 votes

0 answers

99 views

Two-sided estimates of fundamental solutions of second-order parabolic equations

asked Apr 29 at 17:07

0 votes

1 answer

76 views

Decay rate of $\lim_{|z| \to \infty} \|1_{B(z, 1)} f\|_{L^p} =0$

asked Mar 4 at 21:36

0 votes

1 answer

254 views

Is the space $L^p_{\text{loc}} (\mathbb R^d)$ separable w.r.t. the norm $\|f\|_{\tilde L^p} := \sup_{x \in \mathbb R^d} \|1_{B(x, 1)} f\|_{L^p}$?

asked Aug 10, 2023 at 12:01

0 votes

2 answers

125 views

Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable?

asked Aug 15, 2023 at 16:44

0 votes

1 answer

165 views

For $\mu$-a.e. $x \in X$, the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^1 (Y)$. Then $(f_n)$ is Cauchy in $L^1 (X \times Y)$

asked Aug 15, 2023 at 23:37

0 votes

2 answers

107 views

How to construct this sequence that converges a.e. in product measure and that has a very particular form?

asked Aug 17, 2023 at 13:30

0 votes

1 answer

143 views

An estimate of the integral of the higher order derivative of a bump function

asked Aug 25, 2023 at 17:09

0 votes

1 answer

98 views

Does $c$ in the embedding inequality $\|P^\kappa_t \|_{L^p \to L^{p'}} \le c t^{-\frac{d(p'-p)}{2pp'}}$ depend on $\kappa$?

asked Aug 27, 2023 at 13:07

Stack Exchange Network

97 Questions

Does a measurable $F :[0, T] \to L^p (\mathbb R^d; \mathbb R_{\ge 0})$ have a "flattened" measurable version?

Is there $\varepsilon \in (0, 1)$ such that $\sup_{t \in [0, \varepsilon]} [\ell_t]_\beta < \infty$?

Is it true that $\xi \in \partial G (v)$ implies $\frac{\xi}{F'(\phi (v))} \in \partial \phi (v)$?

Is it true that $F(X_0, \cdot) = X_0 + \int_0^T \sigma(s, X_0) \, \mathrm d B_s$ a.s.?

Is a $\sigma$-algebra generated by complete independent $\sigma$-algebras also complete?

Let $D$ be the set of those $\omega \in \Omega$ such that $f(\omega, \cdot)$ is $\mu$-integrable. Is $D$ measurable?

Lipschitz approximation of a probability measure with finite $1$-st moment by the ones with finite $p$-th moment

Is the difference between $\alpha$-Hölder constants of $f\rho$ and $g\rho$ controlled by $\|f-g\|_\infty$?

Lower bound the best $\alpha$-Hölder constant of a convolution

Reference request: Hölder regularity of $(1-\Delta)^{\frac{\alpha}{2}}$ for $\alpha >0$

For any $\beta>0$, there is a constant $c>0$ such that $\left\|(1-\Delta)^{\frac{\beta}{2}} f\right\|_{\infty} \leq c\|f\|_{C_b^\beta}$

Upper bound $I (t) := \sup_{x \in \mathbb R^d} \int_{\mathbb R^d} \frac{|x-y|^\alpha}{t^{d/2}} \exp ( - \frac{|\psi(x) - y|^2}{t} ) \, \mathrm d y$

An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y$

Upper bound $I_R := \int_{B_R^c} |x| (P_t \ell_\nu) (x) \, \mathrm d x$ in terms of $R, \nu, t$?

Is the Boltzmann entropy continuous in the supremum norm?

If we don't care about uniqueness, can we relax the coercivity condition in Lax-Milgram theorem?

Can we approximate a Hölder pdf by higher-order Hölder pdf's?

An identity about Bessel potential operators

How is this interpolating curve well-defined in the minimizing movement scheme?

Is the Boltzmann entropy lower semi-continuous in the weak topology induced by $C_b (\mathbb R^d)$?

Can we lower bound this entropy by $\int_{\mathbb R^d} \rho^k (x) \, \mathrm d x$ and $\int_{\mathbb R^d} |x|^2\rho (x) \, \mathrm d x$?

Are there probability densities $\rho, f_n$ such that $\lim_n \frac{[\rho * f_n]_\alpha}{\|\rho * f_n\|_\infty} = \infty$?

Two-sided estimates of fundamental solutions of second-order parabolic equations

Decay rate of $\lim_{|z| \to \infty} \|1_{B(z, 1)} f\|_{L^p} =0$

Is the space $L^p_{\text{loc}} (\mathbb R^d)$ separable w.r.t. the norm $\|f\|_{\tilde L^p} := \sup_{x \in \mathbb R^d} \|1_{B(x, 1)} f\|_{L^p}$?

Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable?

For $\mu$-a.e. $x \in X$, the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^1 (Y)$. Then $(f_n)$ is Cauchy in $L^1 (X \times Y)$

How to construct this sequence that converges a.e. in product measure and that has a very particular form?

An estimate of the integral of the higher order derivative of a bump function

Does $c$ in the embedding inequality $\|P^\kappa_t \|_{L^p \to L^{p'}} \le c t^{-\frac{d(p'-p)}{2pp'}}$ depend on $\kappa$?