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Consider the following problem. Let $E,F,G$ be real or complex Banach spaces, such that $F\subset G$ with continuous embedding. Let $U\subset E$ an open set and $$ f:U\to G $$ an analytic map, such th …

Edit: I have rewritten my answer after learning more details I have been wondering for a while whether there is an interesting answer concerning analytic functions. I have looked at the papers suggest …

I will summarize what has been said in the comments (thanks to Jochen Glueck for all his help). The answer to the question is no, in general. What is going on is actually very simple. Theorem. Let $E, …

Let $\mathcal H^1(\mathbb R^n)$ be the real Hardy space (as in Stein's "Harmonic Analysis", Chapter 3). It is well known that $\mathcal H^1(\mathbb R^n)\subset L^1(\mathbb R^n)$ and its elements have …

Edit. For the sake of improving the quality of the post, I modified the proof to make it work for all $M>n$ after Prof. Tao’s comments (the previous version was admittedly way too loose). In the end I …

An explicit counterexample The answer to your question is: in general, no. Here I will show a counterexample. As in the previous version of the answer, we set $\Omega=\mathbb R^2$, and for the sake of …

In what follows I will show that the closure of $\ell^1$ under the norm $|x|=\int_1^2|x|_pdp$ is nothing but the Orlicz space $L_\Phi$, where $\Phi$ is the function $$\Phi(t)=\frac{t^2-|t|}{\ln|t|}$$ …

Here is one fairly recent paper of Sebastian Herr (the second author in the paper you linked) and Timothy Candy in which they use $U^p$ and $V^p$ spaces. During a winter school in Obergurgl in 2022, H …

I have asked this question on MSE, but this is a better place. The heat equation and the heat kernel. Consider the heat equation on $\mathbb R$: $$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&=0\qu …

I have asked the same question on MathSE. I was thinking about the following problem. Problem. Given $\alpha>0$, find all values of $\beta\geq 0$ such that the following estimate is true for all $\va …