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A simply written proof is the following. The interpolation inequality $$\|f\|_{L^{p_\theta}}\leq\|f\|_{L^{p_0}}^{1-\theta}\|f\|_{L^{p_1}}^\theta, $$ $$ \frac{1}{p_\theta}=\frac{1-\theta}{p_0}+\frac{\t …

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 …

The problem with this question, compared to this one of yours, is that the vector field on the right hand side of the ODE is not a non-decreasing function of $g$. If you try to make the example of @fe …

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 $$\Phi(t)=\frac{t^2-|t|}{\ln|t|}$$ …

$\def\R{\mathbb R}$$\def\aha{{1/2}}$$\def\maha{{-1/2}}$ Edit. Now it looks correct. I can prove that $x_t$ grows at most like $t$ for $t\geq 1$, up to a multiplicative constant. For simplicity, assume …

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 …

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 …

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 …

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 …