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Heat semigroup norm between fractional Sobolev and $L^p$ spaces

Suppose $phi$ is a weak solution of $Delta phi = f in mathcal{H}^1$. Then $phiin W^{2,1}$Heat equation and semigroup theory.Sobolev spaces and Holder continuity (or, fractional derivatives and singularities)Can we characterize homogeneous Sobolev spaces by means of Sobolev embedding?Heat semigroup on Morrey spacesQuestion about the proof of General Sobolev Inequality in P.D.E. by EvanPossible error in a proof in Jost's Riemannian Geometry and Geometric AnalysisCalculation/Verification of an integral kernel for $operatorname{e}^{tDelta}(1-Delta)^{-frac{1}{4}}$When $L$ is the generator of an analytic semigroup and $alpha, beta >0$, $(-L)^{-alpha}(-L)^{-beta}= (-L)^{-alpha-beta}$Definitions of fractional Sobolev Spaces

5

2

$begingroup$

What is the actual inequality that holds for the heat semigroup between fractional Sobolev space $W^{2alpha,p}$ and classical Lebesgue space $L^q$?

I am trying to derive an inequality
$$
lvertlvert e^{tDelta}f rvertrvert_{W^{2alpha,p}} leq frac{C}{t^beta}
lvertlvert f rvertrvert_{L^q} $$
but i cannot manage to find the right value of $beta$.

I am trying to write the heat semigroup as a convolution
$$ e^{tDelta}f = K_t*f $$ where $K_t$ is the heat kernel, and use Holder inequality for convolutions, but I am stuck when computing
$$ lvertlvert (I-Delta)^alpha K_t rvertrvert_{L^r} $$ where
$frac{1}{q}+frac{1}{r} = frac{1}{p}+1$.

Any help is appreciated.

functional-analysis semigroup-of-operators fractional-sobolev-spaces

share|cite|improve this question

edited Mar 13 at 15:45

Bernard

123k741117

asked Mar 13 at 15:23

Andrea FuzziAndrea Fuzzi

283

$endgroup$

This question has an open bounty worth +100
reputation from MaoWao ending in 11 hours.

This question has not received enough attention.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you want an estimate for all $t$ or only for small enough?
  $endgroup$
  – Andrew
  Mar 18 at 7:48
  

  
  
  
  

add a comment | 

5

2

$begingroup$

What is the actual inequality that holds for the heat semigroup between fractional Sobolev space $W^{2alpha,p}$ and classical Lebesgue space $L^q$?

I am trying to derive an inequality
$$
lvertlvert e^{tDelta}f rvertrvert_{W^{2alpha,p}} leq frac{C}{t^beta}
lvertlvert f rvertrvert_{L^q} $$
but i cannot manage to find the right value of $beta$.

I am trying to write the heat semigroup as a convolution
$$ e^{tDelta}f = K_t*f $$ where $K_t$ is the heat kernel, and use Holder inequality for convolutions, but I am stuck when computing
$$ lvertlvert (I-Delta)^alpha K_t rvertrvert_{L^r} $$ where
$frac{1}{q}+frac{1}{r} = frac{1}{p}+1$.

Any help is appreciated.

functional-analysis semigroup-of-operators fractional-sobolev-spaces

share|cite|improve this question

edited Mar 13 at 15:45

Bernard

123k741117

asked Mar 13 at 15:23

Andrea FuzziAndrea Fuzzi

283

$endgroup$

This question has an open bounty worth +100
reputation from MaoWao ending in 11 hours.

This question has not received enough attention.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you want an estimate for all $t$ or only for small enough?
  $endgroup$
  – Andrew
  Mar 18 at 7:48
  

  
  
  
  

add a comment | 

$begingroup$

What is the actual inequality that holds for the heat semigroup between fractional Sobolev space $W^{2alpha,p}$ and classical Lebesgue space $L^q$?

I am trying to derive an inequality
$$
lvertlvert e^{tDelta}f rvertrvert_{W^{2alpha,p}} leq frac{C}{t^beta}
lvertlvert f rvertrvert_{L^q} $$
but i cannot manage to find the right value of $beta$.

I am trying to write the heat semigroup as a convolution
$$ e^{tDelta}f = K_t*f $$ where $K_t$ is the heat kernel, and use Holder inequality for convolutions, but I am stuck when computing
$$ lvertlvert (I-Delta)^alpha K_t rvertrvert_{L^r} $$ where
$frac{1}{q}+frac{1}{r} = frac{1}{p}+1$.

Any help is appreciated.

functional-analysis semigroup-of-operators fractional-sobolev-spaces

share|cite|improve this question

edited Mar 13 at 15:45

Bernard

123k741117

asked Mar 13 at 15:23

Andrea FuzziAndrea Fuzzi

283

$endgroup$

share|cite|improve this question

edited Mar 13 at 15:45

Bernard

123k741117

asked Mar 13 at 15:23

Andrea FuzziAndrea Fuzzi

283

share|cite|improve this question

edited Mar 13 at 15:45

Bernard

123k741117

asked Mar 13 at 15:23

Andrea FuzziAndrea Fuzzi

283

edited Mar 13 at 15:45

Bernard

123k741117

edited Mar 13 at 15:45

Bernard

123k741117

asked Mar 13 at 15:23

Andrea FuzziAndrea Fuzzi

283

asked Mar 13 at 15:23

Andrea FuzziAndrea Fuzzi

283