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edited Apr 10, 2018 at 12:53

I encountered the following space as a natural space for setting up a certain problem: $$ S_m^p = \{f \colon I \to \mathbb{R} \text{ measurable }; m^{f} \in L^p(I)\} $$ Here, $I$ is an open bounded interval and $m$ is a positive real number. The notation $S_m^p$ is ad hoc.

Is there any other characterization of the space $S_m^p$? What about of $$ S^p = \bigcap_{m \in \mathbb{R}_+} S^p_m? $$

Some remarks

$S_1^p$ is quite large and not very interesting.
$S_m^p$ is not generally a vector space: $\alpha f \in S_m^p$ if and only if $f \in S^p_{m^\alpha}$, and $f+g \in S_m^p$ if and only if $m^f m^g \in L^p$, and $L^p$ is not generally closed with respect to multiplication.
Clearly $L^\infty \subseteq S^\infty$. For the other direction, consider any $m_+ > 1$ and $m_- < 1$. These give an upper and a lower bound for $f$, respectively. Hence $L^\infty = S^\infty$.
Presumably for $1 \leq m_1 \leq m_2$ we have $S^p_{m_2} \subseteq S^p_{m_1}$ (and for $m_2 \leq m_1 \leq 1$), and for $p_1 \leq p_2$ we have $S^{p_2}_{m} \subseteq S^{p_1}_{m}$.

Is there any other characterization of the space $S_m^p$? What about of $$ S^p = \bigcap_{m \in \mathbb{R}_+} S^p_m? $$

Some remarks

$S_1^p$ is quite large and not very interesting.
$S_m^p$ is not generally a vector space: $\alpha f \in S_m^p$ if and only if $f \in S^p_{m^\alpha}$, and $f+g \in S_m^p$ if and only if $m^f m^g \in L^p$, and $L^p$ is not generally closed with respect to multiplication.
Clearly $L^\infty \subseteq S^\infty$. For the other direction, consider any $m_+ > 1$ and $m_- < 1$. These give an upper and a lower bound for $f$, respectively. Hence $L^\infty = S^\infty$.
Presumably for $1 \leq m_1 \leq m_2$ we have $S^p_{m_2} \subseteq S^p_{m_1}$, and for $p_1 \leq p_2$ we have $S^{p_2}_{m} \subseteq S^{p_1}_{m}$.

Is there any other characterization of the space $S_m^p$? What about of $$ S^p = \bigcap_{m \in \mathbb{R}_+} S^p_m? $$

Some remarks

$S_1^p$ is quite large and not very interesting.
$S_m^p$ is not generally a vector space: $\alpha f \in S_m^p$ if and only if $f \in S^p_{m^\alpha}$, and $f+g \in S_m^p$ if and only if $m^f m^g \in L^p$, and $L^p$ is not generally closed with respect to multiplication.
Clearly $L^\infty \subseteq S^\infty$. For the other direction, consider any $m_+ > 1$ and $m_- < 1$. These give an upper and a lower bound for $f$, respectively. Hence $L^\infty = S^\infty$.
Presumably for $1 \leq m_1 \leq m_2$ we have $S^p_{m_2} \subseteq S^p_{m_1}$ (and for $m_2 \leq m_1 \leq 1$), and for $p_1 \leq p_2$ we have $S^{p_2}_{m} \subseteq S^{p_1}_{m}$.

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asked Apr 10, 2018 at 12:47

Tommi

Logarithm of $L^p$ space

Is there any other characterization of the space $S_m^p$? What about of $$ S^p = \bigcap_{m \in \mathbb{R}_+} S^p_m? $$

Some remarks

$S_1^p$ is quite large and not very interesting.
$S_m^p$ is not generally a vector space: $\alpha f \in S_m^p$ if and only if $f \in S^p_{m^\alpha}$, and $f+g \in S_m^p$ if and only if $m^f m^g \in L^p$, and $L^p$ is not generally closed with respect to multiplication.
Clearly $L^\infty \subseteq S^\infty$. For the other direction, consider any $m_+ > 1$ and $m_- < 1$. These give an upper and a lower bound for $f$, respectively. Hence $L^\infty = S^\infty$.
Presumably for $1 \leq m_1 \leq m_2$ we have $S^p_{m_2} \subseteq S^p_{m_1}$, and for $p_1 \leq p_2$ we have $S^{p_2}_{m} \subseteq S^{p_1}_{m}$.