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Michael Hardy
Michael Hardy
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Let $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have $$ \int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} dxdy\lesssim |\log(\epsilon)| $$$$ \int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} \, dx \, dy\lesssim \left|\log(\epsilon)\right| $$ to hold? Does an assumption like $$\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{|x-y|^{n}}\log(|x-y|)^{\alpha} dx dy < \infty $$$$\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{|x-y|^n}\log(|x-y|)^\alpha \, dx \, dy < \infty $$ suffice? Do we need more?

Let $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have $$ \int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} dxdy\lesssim |\log(\epsilon)| $$ to hold? Does an assumption like $$\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{|x-y|^{n}}\log(|x-y|)^{\alpha} dx dy < \infty $$ suffice? Do we need more?

Let $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have $$ \int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} \, dx \, dy\lesssim \left|\log(\epsilon)\right| $$ to hold? Does an assumption like $$\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{|x-y|^n}\log(|x-y|)^\alpha \, dx \, dy < \infty $$ suffice? Do we need more?

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Riku
Riku
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Let $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have $$ \int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} dxdy\lesssim |\log(\epsilon)| $$ to hold? Does somethingan assumption like $$\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{|x-y|^{n}}\log(|x-y|)^{\alpha} dx dy < \infty $$ suffice? Do we need more?

Let $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have $$ \int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} dxdy\lesssim |\log(\epsilon)| $$ to hold? Does something like $$\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{|x-y|^{n}}\log(|x-y|)^{\alpha} dx dy < \infty $$ suffice? Do we need more?

Let $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have $$ \int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} dxdy\lesssim |\log(\epsilon)| $$ to hold? Does an assumption like $$\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{|x-y|^{n}}\log(|x-y|)^{\alpha} dx dy < \infty $$ suffice? Do we need more?

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Riku
Riku
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Estimate on integral with logarithmic weight

Let $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have $$ \int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} dxdy\lesssim |\log(\epsilon)| $$ to hold? Does something like $$\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{|x-y|^{n}}\log(|x-y|)^{\alpha} dx dy < \infty $$ suffice? Do we need more?