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January 19, 2011

Harmonic Analysis Lecture Notes 7

Filed under: 2011 Harmonic Analysis Course — xuhmath @ 6:50 pm

Weak-{L^p} and Lorentz spaces

For further information on this topic, see Chapter 1 of Grafakos’s text, “Classical Fourier Analysis”.

Definition For {1 \leq p < \infty} and integer {d \geq 1}, we define the weak-{L^p({\mathbb R}^d)} space as the vector space of measurable functions {f} on {{\mathbb R}^d} such that

\displaystyle  	||f||^*_{L^{p,\text{weak}}({\mathbb R}^d)} = \sup_{0<\lambda<\infty} \lambda \cdot \big|\{ x : |f(x)| > \lambda \}\big|^{1/p} < \infty.

Equivalently, the space consists of {f} such that {|\{ |f| > \lambda\}| \lesssim \lambda^{-p}}.

Note: The “{*}” is used in the notation to emphasize that this expression is not a norm (cf discussion on {|| \:\:||^*_{L^{p,q}}}).

For comparison’s sake, note that if {f} is a measurable function on {{\mathbb R}^d}, then

\displaystyle  \begin{array}{rcl}  	||f||_{L^p({\mathbb R}^d)} 	&=& \displaystyle \left( \int_{{\mathbb R}^d} \int_{0 \leq \lambda < |f(x)|} p\lambda^{p-1} \; d\lambda \; dx \right)^{1/p} \\ \\ 	&=& \displaystyle \left(\int_0^\infty \big|\{ |f| > \lambda \} \big| p\lambda^{p-1} \; d\lambda \right)^{1/p} \\ \\ 	&=& p^{1/p} \Big|\Big| \lambda |\{ |f| > \lambda \} |^{1/p} \Big|\Big|_{L^p\big( (0,\infty), \frac{d\lambda}{\lambda} \big)}	 	\end{array}

strongly resembles

\displaystyle  	||f||_{L^{p,\text{weak}}} = \underbrace{p^{1/\infty}}_{=1} \Big|\Big| \lambda |\{ |f| > \lambda\} |^{1/p} \Big|\Big|_{L^\infty\left( (0,\infty), \frac{d\lambda}{\lambda}\right)}

and so suggests the following definition.

Definition For {1 \leq p,q < \infty} and integer {d \geq 1}, we define the Lorentz space {L^{p,q}({\mathbb R}^d)} as the vector space of measurable functions {f} on {{\mathbb R}^d} for which

\displaystyle  	||f||_{L^{p,q}}^* := p^{1/q} \Big|\Big| \lambda |\{ |f| > \lambda \}|^{1/p} \Big|\Big|_{L^q\left( (0,\infty), \frac{d\lambda}{\lambda}\right)}

is finite. For the remainder of this discussion, we will write {L^q\left(\frac{d\lambda}{\lambda}\right)} to mean {L^q\left( (0,\infty), \frac{d\lambda}{\lambda}\right)}.

Remark Indeed, {L^{p,p} = L^p} and {L^{p,\infty} = L^{p,\text{weak}}} by the above discussion.

Again {|| \: \:\:||_{L^{p,q}}^*} is not a norm in general. Nevertheless, for all {a \in {\mathbb C}},

\displaystyle  	||af||^*{L^{p,q}} = \Big|\Big| \lambda \big| \{ |f| > \frac{\lambda}{|a|} \} \big|^{1/p} \Big|\Big|_{L^q\left(\frac{d\lambda}{\lambda}\right)} = |a| \: \big|\big| f \big|\big|^*_{L^{p,q}}

and

\displaystyle  \begin{array}{rcl}  	||f+g||^*_{L^{p,q}} &=& \displaystyle \Big|\Big| \lambda \big|\{ |f+g| > \lambda\} \big|^{1/p} \Big|\Big|_{L^q\left(\frac{d\lambda}{\lambda}\right)}\\ 	&\leq& \displaystyle \Big|\Big| \lambda \left( \big| \{ |f| > \frac{\lambda}{2} \} \big| + \big| \{ |g| > \frac{\lambda}{2} \} \big| \right)^{1/p} \Big|\Big|_{L^q\left(\frac{d\lambda}{\lambda}\right)}\\ \\ 	\scriptsize\begin{bmatrix} \text{concavity of fractional}\\ \text{powers and Minkowski} \end{bmatrix}	&\leq& \displaystyle \Big|\Big| \lambda \big| \{ |f| > \frac{\lambda}{2} \}\big|^{1/p} \Big|\Big|_{L^q\left(\frac{d\lambda}{\lambda}\right)} + \Big|\Big| \lambda \big| \{ |g| > \frac{\lambda}{2} \}\big|^{1/p} \Big|\Big|_{L^q\left(\frac{d\lambda}{\lambda}\right)} \\ \\ 	&\leq& 2 ||f||_{L^{p,q}}^* + 2||g||^*_{L^{p,q}}. 	\end{array}

Thus, {|| \:||_{L^{p,q}}^*} is a quasi-norm. When {p \neq 1}, this quasi-norm is equivalent to an actual norm, {|| f||_{L^{p,q}}^* \lesssim_{p,q} || f||_{L^{p,q}} \lesssim_{p,q} || f||_{L^{p,q}}^*}. When {p=1} and {q \neq 1}, there cannot be a norm that is equivalent to {|| \:||_{L^{p,q}}^*}. Nevertheless, there is a metric which generates the same topology. In either case, we obtain a complete metric space.

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