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4 questions
4
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Norm of convolution operator
By Young's inequality for any $f\in L^p(\mathbf{R})$ the map $T_f:g\mapsto f\star g$ is a continuous operator from $L^q(\mathbf{R})$ to $L^r(\mathbf{R})$ where $1\leq p,q,r\leq \infty$ satisfy $1+\...
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3
votes
1
answer
308
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Approximate square root of Dirac delta function on $\mathrm{SL}_2(\mathbb{R})$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\AdS{AdS}$I hope to find a sequence of complex-valued functions $\{f_i(g)\}$ on the group element $g$ of a locally compact group $\SL(2,\mathbb{R})$ so ...
- 175
3
votes
1
answer
667
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Equivalent action of convolution of directional derivative
I have asked this question a while back on StackExchange but have not received any answer/comment. I received a suggestion to post the same question in here which is more research oriented.
Let $k*f(x)...
- 350
0
votes
1
answer
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Does convolution with $(1+|x|)^{-n}$ define an operator $L^p(\mathbb R^n) \to L^p(\mathbb R^n)$
Suppose that $f : \mathbb R^n \to \mathbb R^n$ is a locally integrable function. I am interested in the integral
$$ x \to \int_{\mathbb R^n} ( 1 + |y| )^{-n} f(x-y) \;dy $$
If the decay of the ...
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