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Riku
Riku
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Let $f$$x \in \mathbb{R}^n$ and $f:\mathbb{R^n} \to \mathbb{R}$ be a non-negative function such that $f(x)=0$. Is it true that (assuming $\alpha,\beta>0$) $$\limsup_{r \to 0} r^{-\alpha \beta}\frac{1}{|B_{r}(x)|}\int_{B_{r}(x)} f(y)^\alpha dy < \infty$$ implies $$|f(z)| \le C|z-x|^\beta,$$ for $z$ close enough to $x$ and for some constant $C>0$?

Let $f$ be a function such that $f(x)=0$. Is it true that (assuming $\alpha,\beta>0$) $$\limsup_{r \to 0} r^{-\alpha \beta}\frac{1}{|B_{r}(x)|}\int_{B_{r}(x)} f(y)^\alpha dy < \infty$$ implies $$|f(z)| \le C|z-x|^\beta,$$ for $z$ close enough to $x$ and for some constant $C>0$?

Let $x \in \mathbb{R}^n$ and $f:\mathbb{R^n} \to \mathbb{R}$ be a non-negative function such that $f(x)=0$. Is it true that (assuming $\alpha,\beta>0$) $$\limsup_{r \to 0} r^{-\alpha \beta}\frac{1}{|B_{r}(x)|}\int_{B_{r}(x)} f(y)^\alpha dy < \infty$$ implies $$|f(z)| \le C|z-x|^\beta,$$ for $z$ close enough to $x$ and for some constant $C>0$?

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Riku
Riku
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Morrey condition (integral condition) and (local) Holder condition

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Riku
Riku
  • 839
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  • 17

Morrey condition and Holder condition

Let $f$ be a function such that $f(x)=0$. Is it true that (assuming $\alpha,\beta>0$) $$\limsup_{r \to 0} r^{-\alpha \beta}\frac{1}{|B_{r}(x)|}\int_{B_{r}(x)} f(y)^\alpha dy < \infty$$ implies $$|f(z)| \le C|z-x|^\beta,$$ for $z$ close enough to $x$ and for some constant $C>0$?