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J. E. Pascoe
J. E. Pascoe
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Let $\mu$ be some positive measure on $\mathbb{R}$. For technical reasons, I would like to know if the limit $$\lim_{p\rightarrow\infty}\frac {\ln \|f\|_{L^p(\mu)}}{\ln p}$$ exists in $[0,\infty]$ for any $f$ (That is, I want the limit to exist, but perhaps not be finite.)

Moreover generally I would like to know if in general, $$\lim_{p\rightarrow\infty}\frac {\frac{d^k}{dp^k} \ln \|f\|_{L^p(\mu)}}{\frac{d^k}{dp^k} \ln p}$$ exists in $[0,\infty]$ for any $f,$$f$ such that $f\in L^p(\mu)$ for all $1\leq p<\infty.$ although(Although I only really need it for $k=2.$) Note that these limits are related by L'Hospital's rule.

Let $\mu$ be some positive measure on $\mathbb{R}$. For technical reasons, I would like to know if the limit $$\lim_{p\rightarrow\infty}\frac {\ln \|f\|_{L^p(\mu)}}{\ln p}$$ exists in $[0,\infty]$ for any $f$ (That is, I want the limit to exist, but perhaps not be finite.)

Moreover generally I would like to know if in general, $$\lim_{p\rightarrow\infty}\frac {\frac{d^k}{dp^k} \ln \|f\|_{L^p(\mu)}}{\frac{d^k}{dp^k} \ln p}$$ exists in $[0,\infty]$ for any $f,$ although I only really need it for $k=2.$

Let $\mu$ be some positive measure on $\mathbb{R}$. For technical reasons, I would like to know if the limit $$\lim_{p\rightarrow\infty}\frac {\ln \|f\|_{L^p(\mu)}}{\ln p}$$ exists in $[0,\infty]$ for any $f$ (That is, I want the limit to exist, but perhaps not be finite.)

Moreover generally I would like to know if in general, $$\lim_{p\rightarrow\infty}\frac {\frac{d^k}{dp^k} \ln \|f\|_{L^p(\mu)}}{\frac{d^k}{dp^k} \ln p}$$ exists in $[0,\infty]$ for any $f$ such that $f\in L^p(\mu)$ for all $1\leq p<\infty.$ (Although I only really need it for $k=2.$) Note that these limits are related by L'Hospital's rule.

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J. E. Pascoe
J. E. Pascoe
  • 1.4k
  • 11
  • 20

Let $\mu$ be some positive measure on $\mathbb{R}$. For technical reasons, I would like to know if the limit $$\lim_{p\rightarrow\infty}\frac {\ln \|f\|_{L^p(\mu)}}{\ln p}$$ exists in $[0,\infty]$ for any $f$ (That is, I want the limit to exist, but perhaps not be finite.)

Moreover generally I would like to know if in general, $$\lim_{p\rightarrow\infty}\frac {\frac{d^k}{dp} \ln \|f\|_{L^p(\mu)}}{\frac{d^k}{dp} \ln p}$$$$\lim_{p\rightarrow\infty}\frac {\frac{d^k}{dp^k} \ln \|f\|_{L^p(\mu)}}{\frac{d^k}{dp^k} \ln p}$$ exists in $[0,\infty]$ for any $f,$ although I only really need it for $k=2.$

Let $\mu$ be some positive measure on $\mathbb{R}$. For technical reasons, I would like to know if the limit $$\lim_{p\rightarrow\infty}\frac {\ln \|f\|_{L^p(\mu)}}{\ln p}$$ exists in $[0,\infty]$ for any $f$ (That is, I want the limit to exist, but perhaps not be finite.)

Moreover generally I would like to know if in general, $$\lim_{p\rightarrow\infty}\frac {\frac{d^k}{dp} \ln \|f\|_{L^p(\mu)}}{\frac{d^k}{dp} \ln p}$$ exists in $[0,\infty]$ for any $f,$ although I only really need it for $k=2.$

Let $\mu$ be some positive measure on $\mathbb{R}$. For technical reasons, I would like to know if the limit $$\lim_{p\rightarrow\infty}\frac {\ln \|f\|_{L^p(\mu)}}{\ln p}$$ exists in $[0,\infty]$ for any $f$ (That is, I want the limit to exist, but perhaps not be finite.)

Moreover generally I would like to know if in general, $$\lim_{p\rightarrow\infty}\frac {\frac{d^k}{dp^k} \ln \|f\|_{L^p(\mu)}}{\frac{d^k}{dp^k} \ln p}$$ exists in $[0,\infty]$ for any $f,$ although I only really need it for $k=2.$

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J. E. Pascoe
J. E. Pascoe
  • 1.4k
  • 11
  • 20

Limit of the logarithm of the $L^p$ norm over the logarithm of $p$ as $p$ goes to infinity

Let $\mu$ be some positive measure on $\mathbb{R}$. For technical reasons, I would like to know if the limit $$\lim_{p\rightarrow\infty}\frac {\ln \|f\|_{L^p(\mu)}}{\ln p}$$ exists in $[0,\infty]$ for any $f$ (That is, I want the limit to exist, but perhaps not be finite.)

Moreover generally I would like to know if in general, $$\lim_{p\rightarrow\infty}\frac {\frac{d^k}{dp} \ln \|f\|_{L^p(\mu)}}{\frac{d^k}{dp} \ln p}$$ exists in $[0,\infty]$ for any $f,$ although I only really need it for $k=2.$