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1 vote
0 answers
47 views

Functional inequality for fractional Laplacian

Let $f$ be a nonnegative function on the $d$-dimensional torus $\mathbb{T}^d$, which you can take to be smooth. Let $\bar{f}:=\int_{\mathbb{T}^d}fdx$. I am interested in whether the following ...
1 vote
1 answer
138 views

Equality of two subharmonic functions

Let $u\leq v$ be two locally bounded subharmonic functions in a domain in $\mathbb{R}^n$. Assume that $u=v$ on a dense subset. Is it true that $u=v$ everywhere?
2 votes
2 answers
620 views

Examples of polar sets

I would like to know some examples of the following polar sets (if they exist): a non trivial uncountable polar set in $\mathbb{R}^{2}$; a polar set in $\mathbb{R}^{2}$ contained in $\mathbb{R}$ with ...