Lp−LqL^p-L^q estimates for maximal operators associated to families of finite type curves

We study the boundedness problem for maximal operators $\mathbb{M}$ associated to averages along families of finite type curves in the plane, defined by $$\mathbb{M}f(x) \, := \, \sup_{1 \leq t \leq 2} \left|\int_{\mathbb{C}} f(x-ty) \, \rho(y) \, d\sigma(y)\right|,$$ where $d\sigma$ denotes the normalised Lebesgue measure over the curves $\mathbb{C}$. Let $\triangle$ be the closed triangle with vertices $P=(\frac{2}{5}, \frac{1}{5}), ~ Q=(\frac{1}{2}, \frac{1}{2}), ~ R=(0, 0).$ In this paper, we prove that for $(\frac{1}{p}, \frac{1}{q}) \in (\triangle \setminus \{P, Q\}) \cap \left\{(\frac{1}{p}, \frac{1}{q}) :q > m \right\}$, there is a constant $B$ such that $\|\mathbb{M}f\|_{L^q(\mathbb{R}^2)} \leq \, B \, \|f\|_{L^p(\mathbb{R}^2)}$. Furthermore, if $m <5,$ then we have $\|\mathbb{M}f\|_{L^{5, \infty}(\mathbb{R}^2)} \leq B \|f\|_{L^{\frac{5}{2} ,1} (\mathbb{R}^2)}.$ We shall also consider a variable coefficient version of maximal theorem and we obtain the $L^p-L^q$ boundedness result for $(\frac{1}{p}, \frac{1}{q}) \in \triangle^{\circ} \cap \left\{(\frac{1}{p}, \frac{1}{q}) :q > m \right\},$ where $\triangle^{\circ}$ is the interior of the triangle with vertices $(0,0), ~(\frac{1}{2}, \frac{1}{2}), ~(\frac{2}{5}, \frac{1}{5}).$ An application is given to obtain $L^p-L^q$ estimates for solution to higher order, strictly hyperbolic pseudo-differential operators.Comment: 16 pages. revised version of the file. Several references have been modified. arXiv admin note: text overlap with arXiv:1510.08649, arXiv:1609.0814

“Virtual Water”: a modern means of resolving water distribution disputes between Israel and the Palestinians

The notion of a water deficit in the Middle East is not a new occurrence. Many generations inhabiting the area have been fighting for control over rivers and lakes in order to gain a constant water supply, as well as an advantageous position over the neighboring countries. After 2000, water, like trees, became a special focus of conflict in the Intifada. The government of Israel and the Palestinian Authority have been involved in many negotiations and confrontations struggling to establish an equal water distribution throughout the region. However, due to the violent history of their relations, the compromise is yet to be set

Water and the Treaty of Peace between Israel and Jordan

The Treaty of Peace between the State of Israel and the Hashemite Kingdom of Jordan was signed at the southern border crossing of Wadi Araba on October 26, 1994. It symbolized one of the world’s most famous water – sharing agreements – in addition to making Jordan the only Arab country after Egypt to normalize relations with Israel. The Jordanian – Israeli Peace Treaty is part of the efforts towards joint management of water resources

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Biculturalism in practice, ‘Te Pounamu’: Integration of a Māori model with traditional clinical assessment processes

Traditional clinical psychological assessment follows a western paradigm of interaction, which relies heavily on verbalisations of internal and external symptoms. This process relies upon the assumption of and acceptance that tangata whaiora (consumer/s) are able to do this. Training encourages a clinical psychologist to take into account differences which may or may not be obvious, e.g., differences in religious beliefs, gender, hearing, or sight. However, as clinicians we often neglect to assess tangata whaiora abilities to verbalise such information. Te Pounamu, as a Māori model demonstrates the underlying principles of Te Tiriti O Waitangi. Providing the best treatment outcomes is about acknowledging all aspects of one’s wellbeing: physical, mental, spiritual, and whānau. Te Pounamu provides a process that acknowledges such aspects in a way of knowing that is healing. From clinical experience, the interactive mode that Te Pounamu utilises has been found efficacious in assisting tangata whaiora to understand how such factors contribute to poor mental health