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Let me denote by $F : HS(X,Y) \to \mathbb{R}$ the map which you defined as $F(A) := \|y-Ax\|_Y^2$. By standard arguments, you indeed have that $F$ is $C^1$ and, for any $A \in HS(X,Y)$, $$ \mathcal{L} …

Problem statement Let $f,g \in C^\omega(X,\mathbb{R})$ be two real analytic functions over a real Banach space $X$. Assume that, for every $n \in \mathbb{N}$, there exists $C_n>0$ and $h_n \in C^\omeg …

It is not true that $W^{\sigma,1} \times C^{0,\sigma} \hookrightarrow W^{\sigma,1}$ with $\Omega = \mathbb{R}^n$. My reference for such questions is Thomas Runst, and Winfried Sickel. Sobolev spaces …

Yes, such an interpretation exists, at least in the following case. Take $p = 2$, $n = 1$ and $\Omega = (0,1)$. Then, for $0 < s < 1$, $$ \| f \|_{H^s(0,1)} \approx \| \partial_t^s f \|_{L^2(0,1)} $$ …

It looks like you are looking for the extreme case of a Gagliardo-Nirenberg interpolation inequality, in the case of an exterior domain (that is, an unbounded domain with compact boundary). Such inequ …

I am trying to find smooth functions $f : \mathbb{R}_+ \to \mathbb{R}_+$ such that the quantity $$\Delta_f(x) := 2f(x)-f(2x)$$ is positive for $x$ large enough and has the greatest asymptotic growth. …

I would say that the book Linear and Quasilinear Parabolic Problems, Volume II: Function Spaces by Herbert Amann qualifies as a modern reference (published in 2019). You can find a version of a Rellic …

I believe this is indeed true, if, as mentioned in my comment, one assumes that $P$ is a contractive projection from $X_0 \to X_0$ and $X_1 \to X_1$ and defines $E_\theta := P X_\theta$ for all $\thet …

Most theoretical papers concerning kernels assume that they are given a positive definite kernel. In this question, we want to show that a specific kernel is positive definite. We are interested in t …