Constants in Hardy-Littlewood inequality for Dini derivatives
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I've been reading through some notes on Terence Tao's blog (https://terrytao.wordpress.com/2010/10/16/245a-notes-5-differentiation-theorems/).
Lemma 56 states that for $F:[a,b]to mathbb{R}$ a continuous, monotonically increasing function that the Dini derivatives satisfy
$m({x in [a,b]: D^+(F)(x)>lambda }) leq frac{F(b)-F(a)}{lambda}$
(where $D^+(F)(x)$ is defined as $limsup_{hto 0^+} frac{F(x+h)-F(x)}{h}$)
It is mentioned that in the case that $F$ is not assumed to be continuous that one needs some constant
$m({x in [a,b]: D^+(F)(x)>lambda}) leq Cfrac{F(b)-F(a)}{lambda}$
One can prove this, for example, for $C=3$ with the Vitali covering lemma.
I'd like to construct a function $F$ which actually requires some $C>1$ for the inequality to hold, but am having some trouble doing so. It seems to me that to alter the value of $D^+(F)$ on a measurable set we need to add discontinuities to our function on some countable dense set. For example, we could pick some enumeration of the rationals between $[a,b]$ and alter a function $F$ by adding a jump of size $frac{1}{2^n}$ at each such rational. However, doing any calculations with this new function seems very difficult to me. Does anyone know of any monotonic functions $F$ which require a constant $C>1$ for the above inequality to hold?
real-analysis
asked Jan 19 at 4:07
wfawwerwfawwer
414
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add a comment |
$begingroup$
I've been reading through some notes on Terence Tao's blog (https://terrytao.wordpress.com/2010/10/16/245a-notes-5-differentiation-theorems/).
Lemma 56 states that for $F:[a,b]to mathbb{R}$ a continuous, monotonically increasing function that the Dini derivatives satisfy
$m({x in [a,b]: D^+(F)(x)>lambda }) leq frac{F(b)-F(a)}{lambda}$
(where $D^+(F)(x)$ is defined as $limsup_{hto 0^+} frac{F(x+h)-F(x)}{h}$)
It is mentioned that in the case that $F$ is not assumed to be continuous that one needs some constant
$m({x in [a,b]: D^+(F)(x)>lambda}) leq Cfrac{F(b)-F(a)}{lambda}$
One can prove this, for example, for $C=3$ with the Vitali covering lemma.
I'd like to construct a function $F$ which actually requires some $C>1$ for the inequality to hold, but am having some trouble doing so. It seems to me that to alter the value of $D^+(F)$ on a measurable set we need to add discontinuities to our function on some countable dense set. For example, we could pick some enumeration of the rationals between $[a,b]$ and alter a function $F$ by adding a jump of size $frac{1}{2^n}$ at each such rational. However, doing any calculations with this new function seems very difficult to me. Does anyone know of any monotonic functions $F$ which require a constant $C>1$ for the above inequality to hold?
real-analysis
asked Jan 19 at 4:07
wfawwerwfawwer
414
$endgroup$
add a comment |
$begingroup$
I've been reading through some notes on Terence Tao's blog (https://terrytao.wordpress.com/2010/10/16/245a-notes-5-differentiation-theorems/).
Lemma 56 states that for $F:[a,b]to mathbb{R}$ a continuous, monotonically increasing function that the Dini derivatives satisfy
$m({x in [a,b]: D^+(F)(x)>lambda }) leq frac{F(b)-F(a)}{lambda}$
(where $D^+(F)(x)$ is defined as $limsup_{hto 0^+} frac{F(x+h)-F(x)}{h}$)
It is mentioned that in the case that $F$ is not assumed to be continuous that one needs some constant
$m({x in [a,b]: D^+(F)(x)>lambda}) leq Cfrac{F(b)-F(a)}{lambda}$
One can prove this, for example, for $C=3$ with the Vitali covering lemma.
I'd like to construct a function $F$ which actually requires some $C>1$ for the inequality to hold, but am having some trouble doing so. It seems to me that to alter the value of $D^+(F)$ on a measurable set we need to add discontinuities to our function on some countable dense set. For example, we could pick some enumeration of the rationals between $[a,b]$ and alter a function $F$ by adding a jump of size $frac{1}{2^n}$ at each such rational. However, doing any calculations with this new function seems very difficult to me. Does anyone know of any monotonic functions $F$ which require a constant $C>1$ for the above inequality to hold?
real-analysis
asked Jan 19 at 4:07
wfawwerwfawwer
414
$endgroup$
I've been reading through some notes on Terence Tao's blog (https://terrytao.wordpress.com/2010/10/16/245a-notes-5-differentiation-theorems/).
Lemma 56 states that for $F:[a,b]to mathbb{R}$ a continuous, monotonically increasing function that the Dini derivatives satisfy
$m({x in [a,b]: D^+(F)(x)>lambda }) leq frac{F(b)-F(a)}{lambda}$
(where $D^+(F)(x)$ is defined as $limsup_{hto 0^+} frac{F(x+h)-F(x)}{h}$)
It is mentioned that in the case that $F$ is not assumed to be continuous that one needs some constant
$m({x in [a,b]: D^+(F)(x)>lambda}) leq Cfrac{F(b)-F(a)}{lambda}$
One can prove this, for example, for $C=3$ with the Vitali covering lemma.
I'd like to construct a function $F$ which actually requires some $C>1$ for the inequality to hold, but am having some trouble doing so. It seems to me that to alter the value of $D^+(F)$ on a measurable set we need to add discontinuities to our function on some countable dense set. For example, we could pick some enumeration of the rationals between $[a,b]$ and alter a function $F$ by adding a jump of size $frac{1}{2^n}$ at each such rational. However, doing any calculations with this new function seems very difficult to me. Does anyone know of any monotonic functions $F$ which require a constant $C>1$ for the above inequality to hold?
real-analysis
real-analysis
asked Jan 19 at 4:07
wfawwerwfawwer
414
asked Jan 19 at 4:07
wfawwerwfawwer
414
asked Jan 19 at 4:07
wfawwerwfawwer
414
asked Jan 19 at 4:07
wfawwerwfawwer
414
asked Jan 19 at 4:07
wfawwerwfawwer
414
414
add a comment |
add a comment |
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