University of Connecticut

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Analysis and Probability Seminar
Translation invariant operators in $$L^p$$
Piotr Hajlasz (University of Pittsburgh)

Tuesday, April 16, 2019
2:00pm – 3:00pm

Storrs Campus
MONT 313

We say that a bounded linear operator $$T:L^p(\mathbb{R}^n)\to L^q(\mathbb{R}^n)$$ is translation invariant if $$T(\tau_y f)=\tau_y(Tf)$$ for all $$f\in L^p(\mathbb{R}^n)$$ and all $$y\in\mathbb{R}^n$$, where $$(\tau_y f)(x)=f(x+y)$$. The following result of Hormander plays a fundamental role in harmonic analysis since it applies to all convolution type operators.

Theorem (Hormander 1960). If $$T:L^p(\mathbb{R}^n)\to L^q(\mathbb{R}^n)$$, $$1\leq p<\infty$$, $$1\leq q\leq\infty$$ is non-zero and translation invariant, then $$q\geq p$$.

The proof is simple and well known. The argument does not generalize to the case of $$p=\infty$$. However, the argument still works if we replace $$L^\infty$$ by $$L^\infty_0$$ which is the subspace of $$L^\infty$$ consisting of functions that converge to 0 at infinity. In that case Hormander proved the following result:

Theorem (Hormander 1960). If $$T:L^\infty_0(\mathbb{R}^n)\to L^q(\mathbb{R}^n)$$ is non-zero and translation invariant, then $$q=\infty$$.

Hormander calls this result "somewhat incomplete for $$p=\infty$$." However, the case of $$p=\infty$$ has been completely solved by Liu and van Rooij in a paper that is completely unknown (has only one citation in MathSciNet). Their beautiful and surprising result states as follows:

Theorem (Liu and van Rooij 1974). If $$T:L^\infty(\mathbb{R}^n)\to L^q(\mathbb{R}^n)$$ is non-zero and translation invariant, then $$q\geq 2$$. On the other hand, there is a non-zero translation invariant operator $$T_1:L^\infty(\mathbb{R}^n)\to L^2(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)$$. It follows that $$T_1:L^2(\mathbb{R}^n)\to L^q(\mathbb{R}^n)$$ is bounded for all $$2\leq q\leq\infty$$.

In this talk I will sketch a new proof of this result (joint work with Bownik, Nazarov and Wojtaszczyk).

Contact:

Scott Zimmerman, scott.zimmerman@uconn.edu

Analysis and Probability Seminar (primary), UConn Master Calendar

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