Question

Does $f$ belong to $W^{1,p}(\mathbf{R})$?

score 36 · Answer 1 · 2026-04-28 15:32:34

36

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Does $f$ belong to $W^{1,p}(\mathbf{R})$?

Published on 28 Apr 2026 - 15:32

score 429 · Answer 2 · 2026-04-14 04:25:29

429

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Equivalent norms in polynomial space

Published on 14 Apr 2026 - 4:25

score 620 · Answer 3 · 2026-04-28 16:49:36

620

Views

class of compact support functions is dense in $W^{k,p}(\mathbb R^n)$

Published on 28 Apr 2026 - 16:49

score 68 · Answer 4 · 2026-04-16 01:52:29

Show that $\lVert f_{k}\rVert_{p}\longrightarrow\lVert f\rVert_{p}$ whenever $\lVert f-f_{k}\rVert_{p}\longrightarrow 0$ as $k\longrightarrow\infty$

score 414 · Answer 5 · 2026-04-17 05:13:17

414

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Bounded sequence in $L^1(\omega)$

Published on 17 Apr 2026 - 5:13

score 129 · Answer 6 · 2026-04-09 09:14:11

129

Views

Problem about convergence of operator norm and compactness of an operator.

Published on 09 Apr 2026 - 9:14

score 113 · Answer 7 · 2026-04-16 13:21:49

113

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Prove the Part (a) and Part (b).

Published on 16 Apr 2026 - 13:21

score 418 · Answer 8 · 2026-03-25 07:44:54

418

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Generalization of Lp; $L_{\phi}(\mu)=\{f: \exists M>0 \int \phi(\frac{f(x)}{M})d\mu < +\infty\}$

Published on 25 Mar 2026 - 7:44

score 146 · Answer 9 · 2026-04-17 06:32:21

For each $p\in[1,+\infty)$ , we have $\lVert k_{\varepsilon}*f\rVert_{L^{p}({\bf R}^{n}~)}\longrightarrow0$ as $\varepsilon\longrightarrow0$

score 179 · Answer 10 · 2026-04-28 18:04:34

179

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Weak convergence in Sobolev spaces of $\phi(x+n)$

Published on 28 Apr 2026 - 18:04

score 42 · Answer 11 · 2026-04-09 12:40:08

42

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we define $Ax=\{ \eta_i \}_{i=1}^{\infty},$ where $\eta_i=\sum_{j=1}^{\infty} a_{ij}\xi_j, \ (i=1,2,\ldots),$ Compute $\|A \|$.

Published on 09 Apr 2026 - 12:40

score 268 · Answer 12 · 2026-04-16 14:32:16

$\bigg(\int_{0}^{\infty}\bigg(\frac{1}{x}\int_{0}^{x}|f(t)|dt\bigg)^{p}dx\bigg)^{\frac{1}{p}}\le\frac{p}{p-1}\lVert f\rVert_{L^{p}(0,\infty)}$

score 249 · Answer 13 · 2026-04-17 10:07:38

$\bigg(\int_{0}^{\infty}(\int_{0}^{x}|f(t)|dt)^{p}x^{-b-1}dx\bigg)^{1/p}\le p/b\bigg(\int_{0}^{\infty}|f(t)|^{p}t^{p-b-1}dt\bigg)^{1/p}$

score 112 · Answer 14 · 2026-04-17 01:12:19

Check proof that $\int_0^\infty\left(\int_x^\infty|f(t)|dt\right)^px^{b-1}dx\le\left(\frac pb\right)^p\int_0^\infty|f(t)|^pt^{p+b-1}dt$

score 960 · Answer 15 · 2026-04-11 12:42:17

960

Views

A kind of Minkowski inequality for integral

Published on 11 Apr 2026 - 12:42

List Question

Does $f$ belong to $W^{1,p}(\mathbf{R})$?

Equivalent norms in polynomial space

class of compact support functions is dense in $W^{k,p}(\mathbb R^n)$

Show that $\lVert f_{k}\rVert_{p}\longrightarrow\lVert f\rVert_{p}$ whenever $\lVert f-f_{k}\rVert_{p}\longrightarrow 0$ as $k\longrightarrow\infty$

Bounded sequence in $L^1(\omega)$

Problem about convergence of operator norm and compactness of an operator.

Prove the Part (a) and Part (b).

Generalization of Lp; $L_{\phi}(\mu)=\{f: \exists M>0 \int \phi(\frac{f(x)}{M})d\mu < +\infty\}$

For each $p\in[1,+\infty)$ , we have $\lVert k_{\varepsilon}*f\rVert_{L^{p}({\bf R}^{n}~)}\longrightarrow0$ as $\varepsilon\longrightarrow0$

Weak convergence in Sobolev spaces of $\phi(x+n)$

we define $Ax=\{ \eta_i \}_{i=1}^{\infty},$ where $\eta_i=\sum_{j=1}^{\infty} a_{ij}\xi_j, \ (i=1,2,\ldots),$ Compute $\|A \|$.

$\bigg(\int_{0}^{\infty}\bigg(\frac{1}{x}\int_{0}^{x}|f(t)|dt\bigg)^{p}dx\bigg)^{\frac{1}{p}}\le\frac{p}{p-1}\lVert f\rVert_{L^{p}(0,\infty)}$

$\bigg(\int_{0}^{\infty}(\int_{0}^{x}|f(t)|dt)^{p}x^{-b-1}dx\bigg)^{1/p}\le p/b\bigg(\int_{0}^{\infty}|f(t)|^{p}t^{p-b-1}dt\bigg)^{1/p}$

Check proof that $\int_0^\infty\left(\int_x^\infty|f(t)|dt\right)^px^{b-1}dx\le\left(\frac pb\right)^p\int_0^\infty|f(t)|^pt^{p+b-1}dt$

A kind of Minkowski inequality for integral

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