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Given any $ g \in L^\infty(\mathbb{R})$ , define the associated multiplier operator $ T_g \colon L^2(\mathbb{R}) \to L^2(\mathbb{R})$ by $ $ \mathcal{F}(T_g f) \ = \ g.\mathcal{F}f $ $ where $ \mathcal{F}$ denotes the Fourier transform. Definition. We will say that a function $ f \in L^\infty(\mathbb{R})$ is nice if for every bounded $ gRead more

Assume $ \mathsf{ZF} + \mathsf{DC}$ . Must there exist an injection from $ \aleph_2$ to $ \mathcal{P}(\mathbb{R})$ ? If not, what is the consistency strength of the nonexistence of such an injection? I hope I’m not missing something obvious. Here are my thoughts so far: There is always a surjection from $ \mathcal{P}(\mathbb{R})$ to $Read more

We are trying to build a discrete model for each SLE (Schramm-Loewner evolution) and one key step is solving the following question: Q: Finding a two-dimensional $ \mathbb{H}$ -conformally invariant (details below) process $ X=(X_{1},X_{2})$ in the upper half-plane such that $ $ c\int^{arg(z)}_{0}sin(\theta)^{\beta}d\theta= P_{z}[(X_{1,T_{\mathbb{H}}},X_{2,T_{\mathbb{H}} })\in \mathbb{R}^{-}],$ $ for constant $ c=1/(\int^{\pi}_{0}sin(\theta)^{\beta}d\theta)$ and arbitrary $Read more

I’m trying to prove that there is a constant $ \space C>0 \space$ such that for any function $ \space u \in C^{\infty}_{0} \left ( \mathbb{R} \right ) \space$ following inequality holds: $ {\left \| u \right \|}_{L^2 \left ( \mathbb{R} \right )} \leq C {\left \| u’ \right \|}^{1/2}_{L^1 \left ( \mathbb{R} \right )}Read more